Fixed Income
Securities
VALUATION, RISK, AND RISK MANAGEMENT
Pietro Veronesi
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FIXED INCOME SECURITIES
Valuation, Risk, and Risk Management
Pietro Veronesi
University of Chicago
JOHN WILEY & SONS, INC.
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Copyright 2010
by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada.
c 1994
“P&G Sues Bankers Trust Over Swap Deal” from The New York Times, October 28, 1994 The New York Times. All rights reserved. Used by permission and protected by the Copyright Laws
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Library of Congress Cataloging-in-Publication Data:
Veronesi, Pietro.
Fixed income securities :valuation, risk, and risk management / Pietro Veronesi.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-10910-6 (cloth)
1. Fixed-income securities. 2. Risk management. I. Title.
HG4650.V47 2010
658.15 5–dc22
2009043712
10 9 8 7 6 5 4 3 2 1
To Tommaso
Gabriele,
Sofia,
and Micaela
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CONTENTS
Preface
Acknowledgments
xix
xxxiii
PART I BASICS
1
AN INTRODUCTION TO FIXED INCOME MARKETS
1.1
1.2
1.3
1.4
1.5
Introduction
1.1.1
The Complexity of Fixed Income Markets
1.1.2
No Arbitrage and the Law of One Price
The Government Debt Markets
1.2.1
Zero Coupon Bonds
1.2.2
Floating Rate Coupon Bonds
1.2.3
The Municipal Debt Market
The Money Market
1.3.1
Federal Funds Rate
1.3.2
Eurodollar Rate
1.3.3
LIBOR
The Repo Market
1.4.1
General Collateral Rate and Special Repos
1.4.2
What if the T-bond Is Not Delivered?
The Mortgage Backed Securities Market and Asset-Backed Securities
Market
3
3
6
7
9
11
11
14
14
14
14
14
15
16
18
21
v
vi
CONTENTS
1.6
1.7
1.8
2
The Derivatives Market
1.6.1
Swaps
1.6.2
Futures and Forwards
1.6.3
Options
Roadmap of Future Chapters
Summary
23
23
25
25
26
28
BASICS OF FIXED INCOME SECURITIES
29
2.1
29
30
31
32
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Discount Factors
2.1.1
Discount Factors across Maturities
2.1.2
Discount Factors over Time
Interest Rates
2.2.1
Discount Factors, Interest Rates, and Compounding
Frequencies
2.2.2
The Relation between Discounts Factors and
Interest Rates
The Term Structure of Interest Rates
2.3.1
The Term Structure of Interest Rates over Time
Coupon Bonds
2.4.1
From Zero Coupon Bonds to Coupon Bonds
2.4.2
From Coupon Bonds to Zero Coupon Bonds
2.4.3
Expected Return and the Yield to Maturity
2.4.4
Quoting Conventions
Floating Rate Bonds
2.5.1
The Pricing of Floating Rate Bonds
2.5.2
Complications
Summary
Exercises
Case Study: Orange County Inverse Floaters
2.8.1
Decomposing Inverse Floaters into a Portfolio of Basic
Securities
2.8.2
Calculating the Term Structure of Interest Rates from Coupon
Bonds
2.8.3
Calculating the Price of the Inverse Floater
2.8.4
Leveraged Inverse Floaters
Appendix: Extracting the Discount Factors Z(0, T ) from Coupon
Bonds
2.9.1
Bootstrap Again
2.9.2
Regressions
2.9.3
Curve Fitting
2.9.4
Curve Fitting with Splines
34
38
38
40
42
43
45
47
50
52
52
54
57
57
61
61
62
62
64
65
66
67
67
70
CONTENTS
3
BASICS OF INTEREST RATE RISK MANAGEMENT
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4
The Variation in Interest Rates
3.1.1
The Savings and Loan Debacle
3.1.2
The Bankruptcy of Orange County
Duration
3.2.1
Duration of a Zero Coupon Bond
3.2.2
Duration of a Portfolio
3.2.3
Duration of a Coupon Bond
3.2.4
Duration and Average Time of Cash Flow Payments
3.2.5
Properties of Duration
3.2.6
Traditional Definitions of Duration
3.2.7
The Duration of Zero Investment Portfolios: Dollar Duration
3.2.8
Duration and Value-at-Risk
3.2.9
Duration and Expected Shortfall
Interest Rate Risk Management
3.3.1
Cash Flow Matching and Immunization
3.3.2
Immunization versus Simpler Investment Strategies
3.3.3
Why Does the Immunization Strategy Work?
Asset-Liability Management
Summary
Exercises
Case Study: The 1994 Bankruptcy of Orange County
3.7.1
Benchmark: What if Orange County was Invested in Zero
Coupon Bonds Only?
3.7.2
The Risk in Leverage
3.7.3
The Risk in Inverse Floaters
3.7.4
The Risk in Leveraged Inverse Floaters
3.7.5
What Can We Infer about the Orange County Portfolio?
3.7.6
Conclusion
Case Analysis: The Ex-Ante Risk in Orange County’s Portfolio
3.8.1
The Importance of the Sampling Period
3.8.2
Conclusion
Appendix: Expected Shortfall under the Normal Distribution
vii
73
73
75
75
75
77
78
79
80
82
83
84
86
89
90
91
93
96
97
98
99
103
104
105
105
106
107
108
108
109
110
111
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
113
4.1
113
116
118
118
120
121
Convexity
4.1.1
The Convexity of Zero Coupon Bonds
4.1.2
The Convexity of a Portfolio of Securities
4.1.3
The Convexity of a Coupon Bond
4.1.4
Positive Convexity: Good News for Average Returns
4.1.5
A Common Pitfall
viii
CONTENTS
4.2
4.3
4.4
4.5
4.6
5
4.1.6
Convexity and Risk Management
4.1.7
Convexity Trading and the Passage of Time
Slope and Curvature
4.2.1
Implications for Risk Management
4.2.2
Factor Models and Factor Neutrality
4.2.3
Factor Duration
4.2.4
Factor Neutrality
4.2.5
Estimation of the Factor Model
Summary
Exercises
Case Study: Factor Structure in Orange County’s Portfolio
4.5.1
Factor Estimation
4.5.2
Factor Duration of the Orange County Portfolio
4.5.3
The Value-at-Risk of the Orange County Portfolio with
Multiple Factors
Appendix: Principal Component Analysis
4.6.1
Benefits from PCA
4.6.2
The Implementation of PCA
122
126
127
129
130
132
134
136
137
138
142
142
142
144
145
149
150
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
153
5.1
154
157
158
161
162
164
167
169
170
171
171
175
175
176
178
179
181
182
184
184
189
191
5.2
5.3
5.4
5.5
5.6
5.7
5.8
Forward Rates and Forward Discount Factors
5.1.1
Forward Rates by No Arbitrage
5.1.2
The Forward Curve
5.1.3
Extracting the Spot Rate Curve from Forward Rates
Forward Rate Agreements
5.2.1
The Value of a Forward Rate Agreement
Forward Contracts
5.3.1
A No Arbitrage Argument
5.3.2
Forward Contracts on Treasury Bonds
5.3.3
The Value of a Forward Contract
Interest Rate Swaps
5.4.1
The Value of a Swap
5.4.2
The Swap Rate
5.4.3
The Swap Curve
5.4.4
The LIBOR Yield Curve and the Swap Spread
5.4.5
The Forward Swap Contract and the Forward Swap Rate
5.4.6
Payment Frequency and Day Count Conventions
Interest Rate Risk Management using Derivative Securities
Summary
Exercises
Case Study: PiVe Capital Swap Spread Trades
5.8.1
Setting Up the Trade
CONTENTS
5.8.2
5.8.3
5.8.4
6
192
193
196
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
199
6.1
199
200
202
203
205
208
209
213
220
223
223
225
226
233
6.2
6.3
6.4
6.5
7
The Quarterly Cash Flow
Unwinding the Position?
Conclusion
ix
Interest Rate Futures
6.1.1
Standardization
6.1.2
Margins and Mark-to-Market
6.1.3
The Convergence Property of Futures Prices
6.1.4
Futures versus Forwards
6.1.5
Hedging with Futures or Forwards?
Options
6.2.1
Options as Insurance Contracts
6.2.2
Option Strategies
6.2.3
Put-Call Parity
6.2.4
Hedging with Futures or with Options?
Summary
Exercises
Appendix: Liquidity and the LIBOR Curve
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS
RATE
7.1
7.2
7.3
7.4
The Federal Reserve
7.1.1
Monetary Policy, Economic Growth, and Inflation
7.1.2
The Tools of Monetary Policy
7.1.3
The Federal Funds Rate
Predicting the Future Fed Funds Rate
7.2.1
Fed Funds Rate, Inflation and Employment Growth
7.2.2
Long-Term Fed Funds Rate Forecasts
7.2.3
Fed Funds Rate Predictions Using Fed Funds Futures
Understanding the Term Structure of Interest Rates
7.3.1
Why Does the Term Structure Slope up in Average?
7.3.2
The Expectation Hypothesis
7.3.3
Predicting Excess Returns
7.3.4
Conclusion
Coping with Inflation Risk: Treasury Inflation-Protected Securities
7.4.1
TIPS Mechanics
7.4.2
Real Bonds and the Real Term Structure of Interest Rates
7.4.3
Real Bonds and TIPS
7.4.4
Fitting the Real Yield Curve
7.4.5
The Relation between Nominal and Real Rates
239
239
241
242
243
244
244
247
250
254
255
257
259
261
261
264
264
267
267
268
x
CONTENTS
7.5
7.6
7.7
7.8
8
Summary
Exercises
Case Study: Monetary Policy during the Subprime Crisis of 2007 2008
7.7.1
Problems on the Horizon
7.7.2
August 17, 2007: Fed Lowers the Discount Rate
7.7.3
September - December 2007: The Fed Decreases Rates and
Starts TAF
7.7.4
January 2008: The Fed Cuts the Fed Funds Target and
Discount Rates
7.7.5
March 2008: Bearn Stearns Collapses and the Fed Bolsters
Liquidity Support to Primary Dealers
7.7.6
September – October 2008: Fannie Mae, Freddie Mac,
Lehman Brothers, and AIG Collapse
Appendix: Derivation of Expected Return Relation
271
272
275
276
280
280
281
281
282
282
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
285
8.1
285
287
288
289
290
293
294
295
296
297
300
8.2
8.3
8.4
8.5
8.6
8.7
Securitization
8.1.1
The Main Players in the RMBS Market
8.1.2
Private Labels and the 2007 - 2009 Credit Crisis
8.1.3
Default Risk and Prepayment in Agency RMBSs
Mortgages and the Prepayment Option
8.2.1
The Risk in the Prepayment Option
8.2.2
Mortgage Prepayment
Mortgage Backed Securities
8.3.1
Measures of Prepayment Speed
8.3.2
Pass-Through Securities
8.3.3
The Effective Duration of Pass-Through Securities
8.3.4
The Negative Effective Convexity of Pass-Through
Securities
8.3.5
The TBA Market
Collateralized Mortgage Obligations
8.4.1
CMO Sequential Structure
8.4.2
CMO Planned Amortization Class (PAC)
8.4.3
Interest Only and Principal Only Strips.
Summary
Exercises
Case Study: PiVe Investment Group and the Hedging of Pass-Through
Securities
8.7.1
Three Measures of Duration and Convexity
8.7.2
PSA-Adjusted Effective Duration and Convexity
302
305
306
309
310
314
317
318
324
325
325
CONTENTS
8.8
8.7.3
Empirical Estimate of Duration and Convexity
8.7.4
The Hedge Ratio
Appendix: Effective Convexity
PART II
9
TERM STRUCTURE MODELS: TREES
335
9.1
335
338
338
338
340
343
344
344
345
346
347
348
349
349
350
351
352
353
9.3
9.4
9.5
9.6
A one-step interest rate binomial tree
9.1.1
Continuous Compounding
9.1.2
The Binomial Tree for a Two-Period Zero Coupon Bond
No Arbitrage on a Binomial Tree
9.2.1
The Replicating Portfolio Via No Arbitrage
9.2.2
Where Is the Probability p?
Derivative Pricing as Present Discounted Values of Future Cash Flows
9.3.1
Risk Premia in Interest Rate Securities
9.3.2
The Market Price of Interest Rate Risk
9.3.3
An Interest Rate Security Pricing Formula
9.3.4
What If We Do Not Know p?
Risk Neutral Pricing
9.4.1
Risk Neutral Probability
9.4.2
The Price of Interest Rate Securities
9.4.3
Risk Neutral Pricing and Dynamic Replication
9.4.4
Risk Neutral Expectation of Future Interest Rates
Summary
Exercises
MULTI-STEP BINOMIAL TREES
357
10.1
10.2
357
358
359
361
365
365
367
369
372
376
376
10.3
10.4
10.5
10.6
10.7
11
326
328
330
ONE STEP BINOMIAL TREES
9.2
10
xi
A Two-Step Binomial Tree
Risk Neutral Pricing
10.2.1 Risk Neutral Pricing by Backward Induction
10.2.2 Dynamic Replication
Matching the Term Structure
Multi-step Trees
10.4.1 Building a Binomial Tree from Expected Future Rates
10.4.2 Risk Neutral Pricing
Pricing and Risk Assessment: The Spot Rate Duration
Summary
Exercises
RISK NEUTRAL TREES AND DERIVATIVE PRICING
381
11.1
381
381
383
Risk Neutral Trees
11.1.1 The Ho-Lee Model
11.1.2 The Simple Black, Derman, and Toy (BDT) Model
xii
CONTENTS
11.2
11.3
11.4
11.5
11.6
11.7
12
385
386
387
387
387
392
395
397
398
402
404
406
408
413
413
416
AMERICAN OPTIONS
423
12.1
424
427
428
431
431
435
438
440
444
447
450
451
12.2
12.3
12.4
12.5
13
11.1.3 Comparison of the Two Models
11.1.4 Risk Neutral Trees and Future Interest Rates
Using Risk Neutral Trees
11.2.1 Intermediate Cash Flows
11.2.2 Caps and Floors
11.2.3 Swaps
11.2.4 Swaptions
Implied Volatilities and the Black, Derman, and Toy Model
11.3.1 Flat and Forward Implied Volatility
11.3.2 Forward Volatility and the Black, Derman, and Toy Model
Risk Neutral Trees for Futures Prices
11.4.1 Eurodollar Futures
11.4.2 T-Note and T-Bond Futures
Implied Trees: Final Remarks
Summary
Exercises
Callable Bonds
12.1.1 An Application to U.S. Treasury Bonds
12.1.2 The Negative Convexity in Callable Bonds
12.1.3 The Option Adjusted Spread
12.1.4 Dynamic Replication of Callable Bonds
American Swaptions
Mortgages and Residential Mortgage Backed Securities
12.3.1 Mortgages and the Prepayment Option
12.3.2 The Pricing of Residential Mortgage Backed Securities
12.3.3 The Spot Rate Duration of MBS
Summary
Exercises
MONTE CARLO SIMULATIONS ON TREES
459
13.1
13.2
459
461
13.3
Monte Carlo Simulations on a One-step Binomial Tree
Monte Carlo Simulations on a Two-step Binomial Tree
13.2.1 Example: Non-Recombining Trees in Asian Interest Rate
Options
13.2.2 Monte Carlo Simulations for Asian Interest Rate Options
Monte Carlo Simulations on Multi-step Binomial Trees
13.3.1 Does This Procedure Work?
13.3.2 Illustrative Example: Long-Term Interest Rate Options
13.3.3 How Many Simulations are Enough?
463
465
466
468
469
472
CONTENTS
13.4
13.5
13.6
13.7
13.8
Pricing Path Dependent Options
13.4.1 Illustrative Example: Long-Term Asian Options
13.4.2 Illustrative Example: Index Amortizing Swaps
Spot Rate Duration by Monte Carlo Simulations
Pricing Residential Mortgage Backed Securities
13.6.1 Simulating the Prepayment Decision
13.6.2 Additional Factors Affecting the Prepayment Decision
13.6.3 Residential Mortgage Backed Securities
13.6.4 Prepayment Models
Summary
Exercises
xiii
473
473
473
481
482
483
484
487
490
490
492
PART III TERM STRUCTURE MODELS: CONTINUOUS TIME
14
INTEREST RATE MODELS IN CONTINUOUS TIME
499
14.1
502
504
505
506
510
515
521
525
526
529
14.2
14.3
14.4
14.5
14.6
14.7
14.8
15
Brownian Motions
14.1.1 Properties of the Brownian Motion
14.1.2 Notation
Differential Equations
Continuous Time Stochastic Processes
Ito’s Lemma
Illustrative Examples
Summary
Exercises
Appendix: Rules of Stochastic Calculus
NO ARBITRAGE AND THE PRICING OF INTEREST RATE
SECURITIES
15.1
15.2
15.3
15.4
Bond Pricing with Deterministic Interest Rate
Interest Rate Security Pricing in the Vasicek Model
15.2.1 The Long / Short Portfolio
15.2.2 The Fundamental Pricing Equation
15.2.3 The Vasicek Bond Pricing Formula
15.2.4 Parameter Estimation
Derivative Security Pricing
15.3.1 Zero Coupon Bond Options
15.3.2 Options on Coupon Bonds
15.3.3 The Three Steps to Derivative Pricing
No Arbitrage Pricing in a General Interest Rate Model
15.4.1 The Cox, Ingersoll, and Ross Model
15.4.2 Bond Prices under the Cox, Ingersoll, and
Ross Model
531
532
535
535
537
538
541
545
545
547
548
549
550
551
xiv
CONTENTS
15.5
15.6
15.7
16
552
554
559
559
560
561
DYNAMIC HEDGING AND RELATIVE VALUE TRADES
563
16.1
16.2
16.3
The Replicating Portfolio
Rebalancing
Application 1: Relative Value Trades on the Yield Curve
16.3.1 Relative Pricing Errors Discovery
16.3.2 Setting Up the Arbitrage Trade
Application 2: Hedging Derivative Exposure
16.4.1 Hedging and Dynamic Replication
16.4.2 Trading on Mispricing and Relative Value Trades
The Theta - Gamma Relation
Summary
Exercises
Case Study: Relative Value Trades on the Yield Curve
16.8.1 Finding the Relative Value Trade
16.8.2 Setting Up the Trade
16.8.3 Does It Work? Simulations
16.8.4 Does It Work? Data
16.8.5 Conclusion
Appendix: Derivation of Delta for Call Options
563
565
570
570
570
572
572
575
575
576
578
579
581
584
585
586
588
590
RISK NEUTRAL PRICING AND MONTE CARLO SIMULATIONS
593
17.1
17.2
17.3
593
594
598
599
599
602
603
606
610
611
613
16.4
16.5
16.6
16.7
16.8
16.9
17
Summary
Exercises
Appendix: Derivations
15.7.1 Derivation of the Pricing Formula in Equation 15.4
15.7.2 The Derivation of the Vasicek Pricing Formula
15.7.3 The CIR Model
17.4
17.5
17.6
17.7
17.8
17.9
Risk Neutral Pricing
Feynman-Kac Theorem
Application of Risk Neutral Pricing: Monte Carlo Simulations
17.3.1 Simulating a Diffusion Process
17.3.2 Simulating the Payoff
17.3.3 Standard Errors
Example: Pricing a Range Floater
Hedging with Monte Carlo Simulations
Convexity by Monte Carlo Simulations
Summary
Exercises
Case Study: Procter & Gamble / Bankers Trust Leveraged
Swap
619
CONTENTS
17.9.1
17.9.2
18
Parameter Estimates
Pricing by Monte Carlo Simulations
621
622
THE RISK AND RETURN OF INTEREST RATE SECURITIES
627
18.1
627
18.2
18.3
18.4
18.5
18.6
18.7
19
xv
Expected Return and the Market Price Risk
18.1.1 The Market Price of Risk in a General Interest
Rate Model
Risk Analysis: Risk Natural Monte Carlo Simulations
18.2.1 Delta Approximation Errors
A Macroeconomic Model of the Term Structure
18.3.1 Market Participants
18.3.2 Equilibrium Nominal Bond Prices
18.3.3 Conclusion
Case Analysis: The Risk in the P&G Leveraged Swap
Summary
Exercises
Appendix: Proof of Pricing Formula in Macroeconomic Model
631
631
633
635
636
639
642
644
648
648
649
NO ARBITRAGE MODELS AND STANDARD DERIVATIVES
651
19.1
19.2
651
653
656
658
659
660
663
663
665
669
673
675
675
675
677
677
678
679
681
681
682
682
19.3
19.4
19.5
19.6
19.7
19.8
19.9
No Arbitrage Models
The Ho-Lee Model Revisited
19.2.1 Consistent Derivative Pricing
19.2.2 The Term Structure of Volatility in the Ho-Lee Model
The Hull-White Model
19.3.1 The Option Price
Standard Derivatives under the “Normal” Model
19.4.1 Options on Coupon Bonds
19.4.2 Caps and Floors
19.4.3 Caps and Floors Implied Volatility
19.4.4 European Swaptions
19.4.5 Swaptions’ Implied Volatility
The “Lognormal” Model
19.5.1 The Black, Derman, and Toy Model
19.5.2 The Black and Karasinski Model
Generalized Affine Term Structure Models
Summary
Exercises
Appendix: Proofs
19.9.1 Proof of the Ho-Lee Pricing Formula
19.9.2 Proof of the Expression in Equation 19.13
19.9.3 Proof of the Hull-White Pricing Formula
xvi
CONTENTS
19.9.4
19.9.5
20
683
683
THE MARKET MODEL FOR STANDARD DERIVATIVES
685
20.1
686
688
690
695
20.2
20.3
20.4
21
Proof of the Expression in Equation 19.28
Proof of the Expressions in Equations 19.41 and 19.42
The Black Formula for Caps and Floors Pricing
20.1.1 Flat and Forward Volatilities
20.1.2 Extracting Forward Volatilities from Flat Volatilities
20.1.3 The Behavior of the Implied Forward Volatility
20.1.4 Forward Volatilities and the Black, Derman, and Toy
Model
The Black Formula for Swaption Pricing
Summary
Exercises
FORWARD RISK NEUTRAL PRICING AND THE LIBOR MARKET
MODEL
21.1
21.2
One Difficulty with Risk Neutral Pricing
Change of Numeraire and the Forward Risk Neutral Dynamics
21.2.1 Two Important Results
21.2.2 Generalizations
21.3 The Option Pricing Formula in “Normal” Models
21.4 The LIBOR Market Model
21.4.1 The Black Formula for Caps and Floors
21.4.2 Valuing Fixed Income Securities that Depend on a Single
LIBOR Rate
21.4.3 The LIBOR Market Model for More Complex Securities
21.4.4 Extracting the Volatility of Forward Rates from Caplets’
Forward Volatilities
21.4.5 Pricing Fixed Income Securities by Monte Carlo Simulations
21.5 Forward Risk Neutral Pricing and the Black Formula for Swaptions
21.5.1 Remarks: Forward Risk Neutral Pricing and No Arbitrage
21.6 The Heath, Jarrow, and Morton Framework
21.6.1 Futures and Forwards
21.7 Unnatural Lag and Convexity Adjustment
21.7.1 Unnatural Lag and Convexity
21.7.2 A Convexity Adjustment
21.8 Summary
21.9 Exercises
21.10 Appendix: Derivations
21.10.1 Derivation of the Partial Differential Equation in the Forward
Risk Neutral Dynamics
699
699
702
704
707
707
708
710
711
712
714
715
716
718
720
723
727
729
729
731
733
735
736
737
738
740
740
22
CONTENTS
xvii
21.10.2 Derivation of the Call Option Pricing Formula (Equations
21.11)
21.10.3 Derivation of the Formula in Equations 21.27 and 21.31
21.10.4 Derivation of the Formula in Equation 21.21
21.10.5 Derivation of the Formula in Equation 21.37
741
742
743
743
MULTIFACTOR MODELS
745
22.1
22.2
745
747
748
750
754
755
757
760
762
764
768
768
770
771
773
775
777
781
781
783
785
785
787
Multifactor Ito’s Lemma with Independent Factors
No Arbitrage with Independent Factors
22.2.1 A Two-Factor Vasicek Model
22.2.2 A Dynamic Model for the Short and Long Yield
22.2.3 Long-Term Spot Rate Volatility
22.2.4 Options on Zero Coupon Bonds
22.3 Correlated Factors
22.3.1 The Two-Factor Vasicek Model with Correlated Factors
22.3.2 Zero Coupon Bond Options
22.3.3 The Two-Factor Hull–White Model
22.4 The Feynman-Kac Theorem
22.4.1 Application: Yield Curve Steepener
22.4.2 Simulating Correlated Brownian Motions
22.5 Forward Risk Neutral Pricing
22.5.1 Application: Options on Coupon Bonds
22.6 The Multifactor LIBOR Market Model
22.6.1 Level, Slope, and Curvature Factors for Forward Rates
22.7 Affine and Quadratic Term Structure Models
22.7.1 Affine Models
22.7.2 Quadratic Models
22.8 Summary
22.9 Exercises
22.10 Appendix
22.10.1 The Coefficients of the Joint Process for Short- and
Long-Term Rates
22.10.2 The Two-Factor Hull-White Model
787
787
References
789
Index
797
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PREFACE
It is now the middle of 2009 and finally this book is completed. It has been very exciting
to write a text on the risk and return of fixed income securities, and their derivatives, in the
middle of what many consider the biggest financial crisis since the Great Depression. In
these three years of work the world of finance changed, as many key players in fixed income
markets either collapsed (e.g., the investment banks Bears Stearns and Lehman Brothers),
have been acquired by the U.S. government (e.g., the two mortgage giant agencies Freddie
Mac and Fannie Mae), or have been acquired by other banks (e.g., the investment bank
Merrill Lynch). In this turmoil, the U.S. government has taken the center stage: On the one
hand, the Federal Reserve decreased its reference short-term interest rate, the Federal Funds
target rate, to almost zero, and acted swiftly to set up lending facilities to provide liquidity
to the financial system. On the other hand, the U.S. Treasury used Congress-approved
funds to bail out a number of financial institutions, while the Federal Deposit Insurance
Corporation (FDIC) extended guarantees on the short-term debt of banks in risk of default.
What will this financial turmoil do to fixed income markets around the world?
While it is still hard to forecast how long the recession will last, a certain fact for now is
that fixed income markets will get bigger. And this for several reasons: First, governments’
debt will expand in the future, as governments across the globe increase their spending to
stimulate demand and jump start their economies.1 To do so, governments will need to
borrow even more than in the past, thereby increasing government debt and thus affecting
1 There
is much disagreement on whether such fiscal stimulus will in fact work. However, there is little doubt that
it will increase government debt.
xix
xx
PREFACE
the size of available fixed income securities. For instance, the U.S. government debt in
marketable securities stood at about $6 trillion at the end of 2008, around 40% of the U.S.
Gross Domestic Product (GDP), and the congressional budget office (CBO) predicted an
additional $1.8 trillion U.S. deficit for 2009. A March 2009 analysis of the CBO about the
President’s budget proposal even predicted an increase in U.S. debt held by the public to
56.8% of GDP by 2009 and up to 80% of GDP by 2019.2
Second, in 2008 the U.S. government took upon its shoulders the two mortgage giants
Freddie Mac and Fannie Mae, and therefore their trillion dollars’ worth of debt can now
be considered as safe (or as risky) as U.S. government securities, further expanding the
effective size of U.S. government debt. The debt securities issued by these mortgage giants
are not as simple as U.S. Treasury securities, as they have a number of additional features,
such as embedded options of various kinds, that make their valuation and risk assessment
difficult. The two agencies need to issue these types of securities to hedge against the
variation in the value of the mortgage backed securities (MBS) that they hold in their
assets, a variation that is mainly due to interest rate fluctuations. Indeed, the three agencies
Fannie Mae, Freddie Mac, and Ginnie Mae hold or guarantee about half of the roughly $9
trillion U.S. mortgage market. This implies that about four to five trillion dollars worth of
mortgage backed securities are now guaranteed by the U.S. government. It is only because
of this guarantee, in fact, that the three agencies have been able to issue mortgage backed
securities since the last quarter of 2008, whereas the private market completely dried up.
Given the sheer size of the MBS market, it is as important as ever to understand the pricing
and hedging of such complex fixed income securities.
More generically, a deep understanding of the forces that affect the valuation, risk, and
return of fixed income securities and their derivatives has never been so important. Not
only will fixed income markets be expanding in the future, as mentioned above, but in the
past two years investors across the world dumped risky securities and purchased safe U.S.
government securities, which pushed their prices up and their yields down. Understanding
the forces that move the term structure of interest rates is important to determining what
will happen to these prices once the crisis is over. For instance, how safe is an investment
in long-term U.S. Treasuries? While the U.S. is (still) very unlikely to default, default
risk is but one of the risks that affect the value of Treasury bonds, and an understanding
of the possible losses from an investment in “safe” Treasury bonds is key, especially in an
environment of low interest rates such as the current one. Indeed, the large expansionary
monetary policy of the Federal Reserve, which was necessary to keep the banking sector
from collapsing due to lack of liquidity, may spur a bout of inflation in the future. Inflation
will affect the rate of return on nominal long-term bonds, and therefore the prices of fixed
income securities will adjust. Is an investment in long-term U.S. Treasury bonds really
safe? What about agency mortgage backed securities, which are guaranteed by the U.S.
government as well? Is such an investment riskier than an investment in Treasury securities?
How can derivatives help hedge risks?
2 See
A Preliminary Analysis of the President’s Budget and an Update of CBO’s Budget and Economic Outlook,
Congressional Budget Office, March 2009.
PREFACE
xxi
About this Book
This book covers “fixed” income securities, their valuation, their risks, and the practice of
risk management. I put quotation marks around the term “fixed” because nowadays most
of the so-called fixed income securities have streams of income payments that are all but
fixed. And it is exactly this fact, that most “fixed” income securities in modern financial
markets actually do not have a “fixed income,” that makes the analysis of these debt
instruments difficult. Let’s put some numbers down to see this more precisely. Consider
once again the U.S. market: As of the end of 2008, the U.S. debt stood at about $6 trillion,
approximately 90% of which is in Treasury securities that indeed have a fixed income,
namely, with constant coupons that are paid over time. However, about 10% of the U.S.
debt is in Treasury Inflation Protected Securities (TIPS), that pay a coupon that is not
fixed at all, but fluctuates together with the realized U.S. inflation rate. These fluctuations
make their valuation harder. On top of the $6 trillion Treasury debt, there is a $9 trillion
mortgage backed securities market, whose securities (e.g., pass throughs, collateralized
debt obligations, and so on) have streams of payments that are not fixed, but depend on
various factors, including interest rates’ fluctuations. In addition, we should add the large
swap market, now the main reference market for fixed income security dealers, which had
a global market value of about $8 trillion in 2008. Once again, swaps do not have fixed
income. And finally, the whole fixed income derivatives market, which includes forwards,
futures, and options, adds a few more trillion dollars.
What keeps these markets together?
The concept that I use throughout this text is that of no arbitrage and the law of one
price, that is, the fact that two securities that have the same cash flows should have the
same price. In well-functioning markets, there shouldn’t be (large) arbitrage profits that
are left on the table, as arbitrageurs would step in and trade them away. It is important to
start from the no arbitrage principle to link all of these markets together. Then, after we
have understood the concept of no arbitrage, we can look back and try to understand why
sometimes apparent arbitrage opportunities seem to appear in the market, in the form of
spreads between securities that look similar. Typically, the answer is risk, that is, it may be
risky to set up an arbitrage strategy and carry it out. The 2007 - 2009 crisis provides in fact
an important example of market disruptions, and this book contains several examples and
case studies discussing the risk and return of setting up and carrying out what appear to be
arbitrage strategies.
Why this Book?
The world of fixed income markets is becoming increasingly more complex, with debt
securities that have the most varied payoff structures, and fixed income derivatives that are
growing in sheer size and complexity. Indeed, in many instances it is no longer clear what a
real “derivative” security is. Typically, we think of a derivative security as a security whose
value can be derived from the value of another, more primitive security from the rules of
no arbitrage. However, when the size of a derivative market becomes larger than the one of
the primitive securities, which price depends on which is not clear at all. The swap market,
for instance, which we still call a derivative market and whose size at the beginning of the
xxii
PREFACE
1990s was negligible, now has a global market value of over $8 trillion, and a notional of
over $350 trillion. While we can think of swaps as derivatives, in the more generic sense as
hedging devices or non-funded financial instruments, their valuation does not derive from
anything in particular, but only from the demand and supply of investors who use them for
their needs to hedge or speculate in interest rates.
As the world of fixed income securities becomes more complex, I believe that anyone
who studies fixed income securities must be exposed more directly to this complexity.
This book provides a thorough discussion of these complex securities, the forces affecting
their prices, their risks, and of the appropriate risk management practices. The idea here,
however, is to provide a methodology, and not a shopping list. I do not go over all of
the possible fixed income securities, structured products, and derivative securities that have
ever been invented. I provide instead examples and methodologies that can be applied quite
universally, once the basic concepts are understood. For this reason, the book is filled with
real-world examples and case studies, as discussed below. End-of-chapter exercises using
real-world data and real-world securities cement the important concepts.
In addition, in modern financial markets, countries’ central banks, such as the Federal
Reserve in the United States, actively intervene in fixed income markets to affect interest
rates in the attempt to spur real growth and keep inflation low. A fixed income book
cannot sidestep the central banks’ influence on fixed income securities. I devote a chapter
to discussing the Federal Reserve system, and the relation among interest rates, the real
economy, and inflation. A large recent advance in academic literature links no arbitrage
models with the activities of central banks, and this is important. Similarly, the academic
literature has uncovered numerous stylized facts about the time variation of yields, which I
also briefly summarize in a chapter. For instance, the old idea that an increasing yield curve
predicts higher future interest rates has been proven false in the data time and again, and
we should teach our students the implications of this empirical evidence. In particular, an
increasing yield curve does not predict future higher rates, but future higher bond returns
(i.e., if anything, lower future rates). That is, the literature has uncovered facts about the
time variation of risk premia, which we should talk about in fixed income books. Without
comprehending why yields move, students cannot have a complete understanding of fixed
income markets.
The book also highlights the fact that most of the analysis of fixed income securities
must rely on some models of the term structure, that is, some particular assumptions
about the movement of yields through time. We use such models to link different types
of instruments by no arbitrage and therefore establish the price of one, perhaps complex,
security by using the price of a more primitive security. Such models are used by market
participants both to design arbitrage strategies in proprietary trading desks, or to value
portfolios of derivatives for trading or accounting purposes, or to determine hedge ratios
for risk management reasons. However, this book aims at clarifying two important issues:
First, models have parameters and parameters need data to be estimated. Thus, the use of
data and computers to determine models’ parameters, and therefore to value fixed income
securities, is just part of the fixed income game. We cannot propose to teach students even
the basics of fixed income markets without a long and careful look at the data, and without
knowing how to use data to fit models.
Second, the book clarifies that models are “just models,” and they are always an incomplete description of a much more complex real world. We will see that different models
may yield different answers about the value of the same derivative security even when
PREFACE
xxiii
using the same data to estimate their parameters. There isn’t one right model. Each model
has pros and cons and there is always a tradeoff between using one or another model. For
instance, some models generate simple pricing formulas for relatively complex securities,
and this simplicity is useful if a trader needs to compute the prices of a large portfolio of
derivatives quickly. However, such models may be too simplistic to design an arbitrage
strategy. More complex models take into account more features of the data, but they are
also harder to implement. Finally, some models may work well in some type of interest
rate environments, while others do not because of assumptions that must be made. In this
book, we cover several models, and we go over their properties, the approximations of
reality they make, and their possible drawbacks. The use of examples and case studies, as
well as end-of-chapter exercises enables readers to grasp these differences and understand
why one or another model may be useful in one or another circumstance.
Finally, my aim in writing this book was also to endow anybody who is interested in
fixed income markets, even readers without a strong analytical background, to understand
the complexities, the risks, and the risk management methodologies of real-world fixed
income markets. With this desire in mind, I wrote the book in a way to cover all of the
important concepts in each part of the book, as each part may require a different level of
mathematical sophistication. Parts I and II of the book are accessible to students familiar
with basic calculus, while Part III requires a more analytical background. Still, as discussed
below, Parts I and II are sufficient to cover a complete course in fixed income, and they do
cover all of the deep concepts that I believe anyone who studies fixed income and plays
any role in these markets should possess. The world of fixed income securities has become
more complex, and students who aim at working in this environment must now be able to
recognise and work with this complexity.
I now describe the three parts of the book in more detail.
Part I: Basics
Part I of the book, Chapters 1 to 8, covers the basics of fixed income pricing, risk, and
risk management. After introducing the main fixed income markets in Chapter 1, Chapter
2 contains the building blocks of fixed income relations, namely, the notion of discounts,
interest rates, and the term structure of interest rates. The chapter also discusses the basic
bond pricing formula, as well as some important methodologies for extracting discounts
from observable bond prices. A case study at the end of the chapter further illustrates these
concepts within the pricing of inverse floaters, which are popular fixed income securities
yielding higher-than-market returns if interest rates decline.
Chapter 3 contains the basics of risk management: The chapter introduces the concept
of duration, and its use to design effective hedging strategies, as in asset-liability management. The chapter also introduces the popular risk measures of Value-at-Risk and expected
shortfall. The chapter illustrates these concepts with a discussion of the (likely) risks embedded in the portfolio of Orange County, which lost $1.6 billion and declared bankruptcy
in 1994. Chapter 4 contains some refinements in the risk management techniques introduced in Chapter 3: In particular, the chapter illustrates the notion of bond convexity, and
its implication for risk and risk management, as well as the concepts of yield curve’s slope
and curvature dynamics. This chapter shows that the notion of duration is an incomplete
xxiv
PREFACE
measure of risk, and relatively simple modifications to the model allow for much better
hedging performances, especially through the notion of factor neutrality.
Chapter 5 introduces basic interest rate derivatives, such as forwards and swaps. Besides
describing their properties and their pricing methodology, several examples throughout the
chapter also illustrate the use of such derivative contracts for an effective risk management
strategy. The chapter ends with a case study discussing the risks embedded in a popular
trade, a swap spread trade, a case that also provides further understanding of the swap market
itself. Chapter 6 is the second introductory chapter on derivative securities, covering futures
and options. In particular, the chapter illustrates the notion of options as financial insurance
contracts, which pay only if some particular event takes place. After the description of
futures and options contracts, several examples discuss the usefulness of these contracts
for risk management. In addition, the chapter contains a discussion of the pros and cons of
using forwards, futures, and options for hedging purposes.
A book on fixed income securities must mention the impact that monetary policy has on
interest rates. Chapter 7 discusses the Federal Reserve policy rules, and covers in particular
the Federal Funds rate. A case study at the conclusion of the chapter illustrates the activities
of the Federal Reserve by using the financial crisis of 2007 - 2008 as an example. The
chapter also introduces the Federal funds futures, and the information contained in such
derivative contracts to predict future movements in the Federal funds rate. This chapter also
connects the movement of interest rates over time to real economic growth and inflation
rate, as the Federal Reserve acts to keep the economy growing and the inflation rate low. As
the focus is on inflation, this chapter also covers the Treasury Inflation Protected Securities
(TIPS), Treasury securities that pay coupons and principal that are linked to the realized
inflation rate. Finally, this chapter contains the academic evidence about the variation over
time of interest rates, and the fact that risk premia to hold bonds are time varying. In
particular, this chapter answers the question of why the term structure of interest rates, on
average, slopes upward.
The final chapter of Part I is Chapter 8, which contains a discussion of the mortgage
backed securities (MBS) market, its main players, and the securitization process. Given
that the financial market turmoil of 2007 - 2008 started in the mortgage backed securities
markets, the chapter also describes some of the events during this period of time. This
introductory chapter to mortgage backed securities also contains a discussion of the main
measures of prepayment speed, as well as their impact on the pricing and risk exposure
of several MBS, such as simple pass throughs, collateralized mortgage obligations, and
principal only and interest only strips. The concept of negative convexity is thoroughly
discussed, and illustrated by using data from the main trading market of agency pass
throughs, the To-Be-Announced (TBA) market. A case study at the end of the chapter
also demonstrates how we can measure the duration and convexity of MBSs (and other
securities) by using data instead of pricing formulas.
Part II: Binomial Trees
The second part of the book introduces readers to the concept of term structure modeling
and no arbitrage strategies. Chapter 9 illustrates these important concepts in the simple
framework of one-step binomial trees. I use this chapter to discuss both the relative pricing
of different fixed income instruments, the notion of risk premium of a fixed income security,
as well as the popular pricing methodology called risk neutral pricing. The chapter does
PREFACE
xxv
not use any more mathematics than Part I does, but it is the first step into a bigger world,
the world of no arbitrage term structure models. Chapter 10 extends the analysis to multistep trees. Students will learn the concepts of dynamic replication and hedging. These
are strategies that allow a trader to hedge a contingent payoff in the future by using a
portfolio of other fixed income securities, and understanding them is at the heart of no
arbitrage pricing. The chapter also discusses a simple methodology to build long-term
trees from the prediction of future short-term interest rates, as well as the concept of
risk adjusted probabilities and risk premia. Real-world examples including the pricing of
long-term structured derivatives illustrate how the methodology can be readily applied to
price relatively complex securities. Finally, the chapter introduces the concept of spot rate
duration, which is a concept of duration analogous to the one introduced in Chapter 3, but
for securities defined on binomial trees.
Chapter 11 applies the concepts described in the previous two chapters to illustrate the no
arbitrage pricing of numerous derivative securities. The chapter uses two popular models,
the Ho-Lee model and Black, Derman, and Toy model, to show the differences in pricing
between different models, even when the inputs are the same. These differences allow me
to describe the various properties of the models. We use these models also to price standard
derivatives, such as caps, floors, swaps and swaptions. In addition, the chapter introduces
the notion of implied volatility, that is, the volatility of interest rates that is implied by
the value of options. Building on these multi-step binomial tree models, Chapter 12
investigates the pricing of American options, that is, options that can be exercised any time
before maturity. Several securities have embedded American options, including callable
bonds and mortgage backed securities. This chapter illustrates the concepts of American
options, and the methodology to price them, by going through several examples, such as
Treasury callable securities, American swaptions, and mortgage backed securities. This
chapter also shows the negative convexity that is generated by the American option feature
embedded in such securities.
Finally, Chapter 13 illustrates a new methodology, Monte Carlo simulations, to price
very complex securities on binomial trees. There are securities that cannot be easily priced
on binomial trees because their payoff at maturity may depend on a particular path of
interest rates. However, we can use computers to simulate interest paths on the tree itself,
and therefore obtain the prices and hedge ratios of these securities by simulation. The
chapter applies the methodology to relatively complicated real-world securities, such as
amortizing swaps and mortgage backed securities.
Part III: Continuous Time Models
Part III covers more advanced term structure models that rely on continuous time mathematics. While this part is self contained, as it contains all of the important mathematical
concepts, readers should be ready to see a substantial step up in the analytical requirement
compared to the previous two parts of the book, which, as mentioned, instead only require
a background in basic calculus.
Chapter 14 introduces the notions of Brownian motion, differential equations and Ito’s
lemma. I introduce the concept of a Brownian motion by relying on the intuition developed
on binomial trees, namely, as a limiting behavior of rates as the time-step in the binomial
tree converges to zero. Differential equations are introduced only through examples, as
my aim here is to provide students with the notion of differential equations, and not the
xxvi
PREFACE
methodology to solve for them. I also illustrate the concept of Ito’s lemma by relying on
the convexity concepts discussed earlier in Chapter 4. I apply the concepts of Brownian
motions and Ito’s lemma in Chapter 15 to illustrate the notion of no arbitrage, and obtain
the fundamental pricing equation, an equation that we can use to compute the price of any
fixed income derivative. I focus on the Vasicek model, a model that is relatively simple but
also realistic, and provide several examples on the pricing of real-world securities. In this
chapter I tackle the issue of how to estimate the model’s parameters, and show the potential
shortcomings of the model. The chapter also illustrates the use of this model for the pricing
of options.
Chapter 16 takes the model one step further, and discusses the issue of dynamic rebalancing and relative value trades. Essentially, all fixed income securities are linked to
each other by the variation of interest rates, and therefore they move in a highly correlated
fashion. An interest rate model allows us to compute the price of one security by using
a portfolio of other securities, so long as the latter is properly rebalanced over time as
interest rates change. The methodology is illustrated through various real-world examples,
as well as a case study at the end of the chapter which features real data, and demonstrates
the methodology in action. The chapter also illustrates some drawbacks of using simple
models.
Chapter 17 introduces the second important result of continuous time finance, namely,
the Feynman Kac formula, which provides the solution to the fundamental pricing equation
obtained in Chapter 16. This formula is at the basis of the risk neutral pricing methodology
widely used by market participants to price fixed income securities. In addition, this
formula also justifies the use of some type of Monte Carlo simulations to price fixed
income securities. The chapter provides numerous real-world examples, as well as a case
study discussing the fair valuation of the leveraged swap between Bankers Trust and Procter
& Gamble, which was at the center of a famous court case in 1994. Indeed, Chapter 18
covers the topics of risk measurement and risk management within continuous time models:
In particular, I illustrate the notion of market price of risk, the fair compensation that a fixed
income investor should expect to realize when he or she purchases a fixed income security,
as well as Monte Carlo simulations for risk assessment. I illustrate the use of Monte Carlo
simulations for risk assessment both in examples, as well as in a case study at the end of
the chapter. The chapter also includes an economic model of the term structure, which
links the continuous time models illustrated in earlier chapters to the variation in expected
inflation, and the compensation for risk that investors require to hold nominal securities
when there is inflation.
Chapter 19 discusses no arbitrage models, which are models similar to the ones introduced in Chapter 11 on binomial trees, but in continuous time. The inputs of these models
are the bond prices, and the outputs are the prices of derivative securities. The chapter
offers several applications, and further highlights the pros and cons of different types of
models. I carry on this discussion in Chapter 20, which illustrates the Black’s formula to
price standard derivatives, such as caps, floors, and swaptions. This chapter also links back
to Chapter 11 in what concerns the notion of implied volatility. The chapter also discusses
the important concepts of flat and forward volatility, as well as the dynamics of the term
structure of volatility over time. These concepts are so important in modern financial markets that I decided to present this material in isolation from the previous chapters in Part
III, so that the material in this chapter stands alone, and can also be used as a concluding
chapter after Chapter 11.
PREFACE
xxvii
Chapter 21 introduces a more recent pricing methodology, the forward risk neutral pricing methodology, as well as the more recent Heath, Jarrow, and Morton (HJM) model, and
the Brace, Gatarek, and Musiela (BGM) model. Several applications show the usefulness
of these new models to obtaining the price of even more complicated securities, although
often by relying on Monte Carlo simulations.
I conclude this third part of the book, and the book itself, with Chapter 22, which
extends the concepts developed in the previous chapters to the case in which the yield curve
is driven by multiple factors. Luckily, the main concepts developed earlier readily extend
to multifactor models. I show the additional flexibility offered by these multifactor models
to price interesting additional structured notes and derivative securities, such as those that
depend on multiple points of the term structure.
Pedagogical Strategy
This book employs a hands-on strategy to highlight the valuation, the risks, and the risk
management of fixed income securities. The text is filled with real-world examples and
case studies, which I use to show step by step the fair valuation of most securities, the return
an investor should expect from an investment, and the riskiness of such an investment. I
always use data to set up an example or to illustrate a concept, not only because it makes the
lesson more relevant, but because it shows that we can actually tackle real-world valuation
problems by studying the concepts illustrated in each chapter.
Examples
Each chapter contains many numerical examples illustrating the concepts introduced in the
chapter. Sometimes I use examples to motivate new important concepts. As mentioned,
such examples are always based on real data and therefore on real situations. Even so,
examples are stripped down versions of much more complex problems, and I use such
examples to illustrate one issue at a time.
Case Studies
The book contains several end-of-chapter case studies. These case studies apply the
concepts developed in the chapter to more complex real-world situations. Such situations
may involve the pricing of some structured derivatives, or their risk assessment using some
measures of risk, or describe an arbitrage trading situation and the risk involved in carrying
it out. Unlike the examples, which are tightly focused on the particular issue just being
discussed in the chapter, a case study describes a situation and then carries out the whole
analysis, although of course still within the topic discussed in the chapter. I use case studies
also to show that we must often make many approximations when we apply relatively
simple formulas or models to real-world data. That is, the world is much more complicated
than the simple models or formulas would imply.
Not all chapters have case studies, as it depends on the topic of each chapter. If a
chapter is too simple, for instance, because it is only introductory, then it is hard to apply
the concept to a real-world situation, which tends to be complicated.
xxviii
PREFACE
Data
The book relies heavily on real-world securities data. I use data to illustrate the examples
in the body of the textbook as well as to discuss the case studies at the end of chapters.
In addition, most of the exercises require some data analysis. These data are collected in
spreadsheets, which are available with the textbook. The decision to rely foremost on the
use of data as a pedagogical device springs from my beliefs that only by doing the analysis
with real-world numbers can a student really understand not only the concepts illustrated
in the particular chapter, but also the complexity of applying models to the real world.
From the very beginning we will see that it is actually hard to apply the simple formulas
of fixed income models, even the most elementary ones such as a present value formula,
to real-world data. It is important for students to realize this fact early on, and it is this
challenge that makes the study of fixed income markets so fascinating.
Exercises
Each chapter contains several exercises that cover the topics discussed, and highlight
additional features of real-world fixed-income securities or trading methodologies. A
solutions manual is available to instructors. The exercises are an integral part of the
learning strategy: Most exercises are data driven and require the use of computers, either
spreadsheets (for Parts I and II) or a programming software (for Part III). In modern financial
markets computers are just a necessary part of the analysis toolbox. For instance, in Part I
exercises require spreadsheets to compute the prices of complicated securities from simpler
ones, or their duration and convexity. In Part II, the exercises require spreadsheet programs
to build binomial trees that fit real-world fixed income securities, such as bonds, swaps
and options. Moreover, in some chapters, the exercises require students to carry out Monte
Carlo simulations, on the binomial trees, to value real-world fixed income securities with
embedded options, such as the Bermudan callable bonds of Freddie Mac.
In Part III, the exercises again rely on real-world data to fit more complex models of
the term structure, and ask students to price relatively complex securities. In addition,
exercises often require students to carry out a risk analysis, by computing hedge ratios or
risk measures. The hands-on approach will make clear why practitioners use one model or
another in the various circumstances: Students will experience firsthand the difficulties of
dealing with data even when using relatively simple models.
Software
There are numerous examples in the book which use real-world data to illustrate the concepts
discussed in each chapter. Together with the data sets in such examples, I also include all of
the spreadsheets (for Part I and II) or computer codes (for Part III) that generate the results
of the analysis in the numerical examples. These spreadsheets and computer codes should
be used as a guide not only to better understand the examples themselves, but also to carry
out a similar analysis in the end-of-chapter exercises.
PREFACE
xxix
For Instructors
The material in this book can be taught at two different levels: An introductory level and
an advanced (but not very advanced) level.
Course I: Introduction to Fixed Income Securities
Parts I and II introduce basic analytical tools, and students familiar with basic calculus
should be able to follow them relatively easily. This material covers a full semester fixed
income course for both MBA or undergraduate students. Yet, notwithstanding the relative
simplicity of these two parts of the book, the hands-on strategy, the real-world examples,
the case studies, and the focus on real-world securities provide a strong foundation for
the important concepts in fixed income asset pricing, from no arbitrage to risk premia,
from duration to positive and negative convexity, from risk measurement to risk neutral
pricing. Students at the end of the course will have the tools to tackle the proper analysis of
real-world securities, assess their risk, and perform Monte Carlo simulations (on binomial
trees) to value complex securities. These tools are very important to uncover the often
hidden risks in some structured interest rate securities.
Pedagogically, the chapters’ order already offers guidance on how to progress with the
material. Each chapter’s content often contains the seeds of concepts described in future
chapters. For instance, in Chapter 1 I describe the repurchase agreement (repo) market,
because in Chapter 2, which covers the present value formulas and the use of the law of
one price, I can leverage on the repo market to describe how financial institutions actually
carry out long-short strategies. Similarly, in Chapter 2 I describe floating rate notes, not
only because I can then use this concept to illustrate the pricing of inverse floaters (a case
study at the end of the same chapter), but also because in Chapter 5 I use the same concept
to describe the pricing of swaps, which is the largest fixed income market by notional
amount (about $350 trillion at the end of 2008). The chapters are highly interrelated and
cross-reference each other, and therefore I believe it is pedagogically important to move
forward chapter by chapter.
There is one final remark I want to make in regard to an introductory course in fixed
income. Part II of the book, and especially Chapter 11, discusses the pricing and hedging
of plain vanilla derivatives, such as caps, floors, swaps, and swaptions. This chapter also
dicusses the concepts of implied volatility, flat volatility and forward volatility in the context
of two specific models, the Ho-Lee model and the Black, Derman, and Toy model. This
part should therefore be useful to link this material to the notion of implied volatility from
the Black formula, the standard market formula used to quote standard derivatives. With
this link in mind, I wrote Chapter 20 in Part III in a way that does not need any of the more
advanced material in the earlier chapters of Part III. I just introduce the Black formula,
and discuss the dynamics of implied volatilities over time, and the concept of flat and
forward volatility. The formula is as difficult as the Black and Scholes formula for options
on stocks, so depending on how advanced the students are, they may or may not find the
material challenging.
xxx
PREFACE
Course II: An Advanced Course in Fixed Income Securities
The advanced course would make full use of Part II and Part III of the book. This is the
course I regularly teach to the second year MBA students at the University of Chicago
Booth School of Business, and it is also appropriate for students enrolled in master in
finance programs. The prerequisites for my course include an investment course and a
basic options course, although I often allow students with a solid mathematical background
to take the course without the prerequisites. I cover briefly the concepts in Chapters 1
to 6, which serve mainly to set the notation for the course. I then teach both binomial
trees (Part II) and continuous time models (Part III), more or less in the progression
described in the book. Indeed, Chapter 14 refers to Chapter 11 to introduce the notion of
a Brownian motion as the limit of a binomial tree, as the step size becomes infinitesimally
small. The key concepts that are explored in Part II are then also repeated in Part III,
but by using continuous time methodologies. Students find it very useful to see the
same concepts introduced in binomial trees repeated in a continuous time framework, as
their intuition becomes solidified, especially through the plentiful examples. However,
the greater flexibility offered by the continuous time model enables me to discuss many
more models which are not covered in binomial trees, even with many stochastic factors
(in Chapter 22). Students tend to enjoy the comparison across models, and why some
models work in some interest rate environments and not in some others. To this end, I
give my students challenging, data-oriented homeworks to make them aware not only of
the vast possibilities offered by fixed income term structure models, and their usefulness
to price, hedge or implement a risk analysis of a given security, but also to have students
realize the limitations of such fixed income models, and the fact that models need data for
their effective application to the real world. My homework is always based on real-world
securities that need to be priced, hedged, or, more generically, analyzed, and I wrote most
of the end-of-chapter exercises in this book with this aim in mind, namely, to have students
analyze real-world securities by using the models discussed in each chapter. Sometimes
the analysis require students to gather data from other sources available on public Web
sites, such as the LIBOR fixes available at the British Bankers’ Association Web site, or
the swap rates, available at the Federal Reserve Web site. The data analysis is an integral
part of the book and the learning experience. In term of material, finally, my students also
find it useful to connect the economic model discussed in Chapter 18 to the Vasicek model,
discussed earlier, as well as to the evidence on expected return in Chapter 7, as they see
the connections between risk, risk aversion, return, market price of risk, and, ultimately,
pricing.
Conclusion
To conclude this introduction to the book, let me mention that I truly hope that this book
will encourage readers and students to analyze fixed income markets in a very systematic
way, always looking for the reason why some events occur, some trades seem possible, or
some models may or may not work. I hope that my decision to have two full parts of the
book requiring only a minimal analytical background will push readers to try to correctly
assess the riskiness of complex fixed income securities, to see better what they are buying,
and whether there is any reason why a security may appear to yield a higher-than-market
return. Similarly, regulators may use the same tools to assess the fair valuation of complex
PREFACE
xxxi
securities, at least to first order, without needing a Ph.D. in mathematics or physics. In
the same way, nowadays it is much harder to understand how the engine of a car works,
compared to the past, and mechanics need to have a better knowledge about these new
engines, participants in fixed income markets, whether traders, risk managers, regulators
and so on, cannot hope to use old tools to understand modern markets, as their complexity
has just increased through time, and new tools are necessary. I hope this book will provide
the tools, even to the less mathematically oriented reader, to understand the complexities
of fixed income modern markets.
Pietro Veronesi
Chicago
June, 2009
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ACKNOWLEDGMENTS
First of all, I want to thank my students at the University of Chicago Booth School of
Business, whose enthusiasm for fixed income securities convinced me to write this book.
Their feedback on earlier versions of the manuscript was invaluable. I also thank John
Heaton (The University of Chicago Booth School of Business), Jakub Jurek (Princeton
University), Nick Roussanov (The Wharton School, University of Pennsylvania), and
Richard Stanton (The Haas School of Business, University of California at Berkeley) for
being so brave to adopt an early draft of this book in their MBA or Master courses, so that
I could collect very valuable feedback from them and their students. I also would like to
thank Monika Piazzesi (Stanford University) and Jefferson Duarte (Rice University) for
their early feedback, as well as Senay Agca (George Washington University), David T.
Brown (University of Florida), Robert Jennings (University of Indiana), Robert Kieschnick
(University of Texas at Dallas), David P. Simon (Bentley College), Donald J. Smith (Boston
University), Michael Stutzer (University of Colorado), Manuel Tarrazo (University of San
Francisco), and Russ Wermers (University of Maryland) for their comments. Francisco
Javier Madrid and Nina Boyarchenko provided precious help with some exercises and case
studies, and I thank them for this. I also thank Chetan Dandavate and Camilo Echeverri for
pointing out some important typos in the manuscript. I am also indebted to the development
editor, Peggy Monahan-Pashall, who went through the 800 pages of the manuscript, and
not only corrected all my English mistakes, but provided valuable constructive feedback
on the write up itself. I also thank Jennifer Manias, the Associate Editor, for helping with
the logistics of the publication process. Finally, this book exists also because my editor,
Judith Joseph of Wiley, pushed me to write it. Maybe I should have not listened, but it is
too late now.
P. V.
xxxiii
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PART I
FIXED INCOME MARKETS
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CHAPTER 1
AN INTRODUCTION TO FIXED INCOME
MARKETS
1.1
INTRODUCTION
In the past two decades, fixed income markets have experienced an impressive growth, both
in market value and in complexity. In the old days, until the end of the 1980s, fixed income
markets were dominated by government debt securities, such as United States government
Treasury bills, notes, and bonds. These securities were also relatively simple, as the U.S.
government mainly issued bonds paying a fixed amount of money semi-annually. Although
other governments, such as those of the United Kingdom and Italy, also experimented with
other types of debt securities whose semi-annual payments were not fixed, but rather linked
to a floating index, for instance, the inflation rate, such markets were relatively small. Thus,
the U.S. government debt market was the main reference for global fixed income markets.
Today, however, the U.S. government debt is no longer the dominant fixed income
market, not so much because the U.S. debt shrank over the past two decades, but rather
because other fixed income markets rose substantially relative to U.S. debt and became the
main reference for fixed income pricing. Table 1.1 provides a snapshot of the sizes of fixed
income markets as of December 2008. The first block of markets comprises the traditional
fixed income markets, including U.S. government debt securities, municipal bonds, federal
agency securities and the money market. The total size of these debt markets is around $15
trillion. The next block shows the size of the mortgage backed securities and asset-backed
securities markets. In particular, the mortgage backed securities market stands as a $8.9
trillion market, a good $3 trillion larger than U.S. debt.
3
4
AN INTRODUCTION TO FIXED INCOME MARKETS
Table 1.1 The Size of Fixed Income Markets: December 2008
Market
Market Value
(billion of dollars)
U.S. Treasury Debt
U.S. Municipal Debt
U.S. Federal Agency Securities
U.S. Money Market
5,912.2
2,690.1
3,247.4
3,791.1
Mortgage Backed Securities
Asset-Backed Securities
8,897.3
2,671.8
OTC Interest Rate Swaps
OTC Interest Rate Forwards
OTC Interest Rate Options
Exchange Traded Futures
Exchange Traded Options
16,572.85
153.19
1,694.22
U.S. Corporate Debt
Credit Derivatives
6,280.6
5,651
Notional
(billion of dollars)
328,114.49
39,262.24
51,301.37
19,271.05
35,161.34
41,868
Source: Securities Industry and Financial Market Association (SIFMA) and Bank for International Settlements
(BIS).
Similarly, the next block of markets in Table 1.1 shows the interest rate derivatives
markets. Interest rate swaps, in particular, have a market value of $16 trillion, and a
notional value of $328 trillion. Although neither figure can be compared directly to the
U.S. debt market, for a number of reasons discussed in Chapter 5, the sizes of these markets
once again demonstrate that the U.S. debt market has been eclipsed by other types of
securities. In particular, although in the 1980s and 1990s we would think of swaps as
derivative securities, which “derive” their price from the value of primary securities, such
as Treasuries, it is hard to believe that this is still the case now due to its sheer size. To
any extent, we should consider the swap market a primary market whose value is driven
by investors’, speculators’ and end users’ fluctuating demands. Finally, corporate debt has
increased dramatically in the past few years, with a debt value of about $6.2 trillion. Note
too that the growing market of credit derivatives has reached a market value of $5.6 trillion,
and a notional value of $42 trillion.
The changes in these markets are evident also in Figures 1.1 and 1.2. Considering first
the Treasury debt market, we see that from 1986 to 1996 it grew steadily. The economic
expansion that started in 1991, which would end in 2001, also generated a government
surplus between 1996 and 1999, which led the U.S. government to initiate a policy of
debt buyback. This is evident in the decrease in the face value of government debt during
this period. The U.S. debt started growing again in 2001, to reach about $5.9 trillion in
December 2008. The interesting fact about Figure 1.1, though, is the rise of another market,
which has become a dominant market in the U.S., namely, the market of mortgage backed
securities. From its value of only $372 billion in 1985 it increased steadily over time, to
become larger than the U.S. debt market in 1999, and to become $3 trillion larger than
the U.S. debt market by December 2008. The growth in this market is due to the growth
of the U.S. real estate market, which boomed in the 2000s to reach its peak in 2006, as
well as the steady increase in leverage of U.S. households, who had been taking larger
5
INTRODUCTION
Figure 1.1
The Growth in Market Size
9
8
Mortgage
Related
7
Corporate
Trillion of Dollars
6
5
U.S. Treasury
Debt
4
Money
Market
3
2
Federal
Agency
1
Municipal
Asset−Backed
0
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Source: The Securities Industry and Financial Markets Association (SIFMA)
and larger mortgages and home equity loans to finance consumption. The slight decrease
in this market size visible at the end of the sample, in 2008, is indeed a reflection of the
decline in the housing market and the U.S. recession that started in January 2007. Finally,
a similar growth occurred in corporate debt, which next to U.S. debt was comparatively
small in 1985, but grew steadily over the years, to reach $6.2 trillion by December 2008.
Figure 1.2 plots the stunning growth in interest rate derivatives markets. The interest
rate swaps market, which was negligible at the beginning of the 1980s, experienced an
exponential growth, reaching $328 trillion (notional) by December 2008. The figure also
plots the combinations of over-the-counter (OTC) and exchange-traded interest rate options,
which also grew considerably during this time frame, to reach about $100 trillion notional
by December 2007, although it declined to $86 trillion by December 2008, in the midst of
the 2007 - 2009 financial crisis. Similarly, forward rate agreements and futures contracts
also grew over time, although at a much slower rate.
The bottom line of this discussion is that fixed income markets are very large and still
growing. Moreover, there is not a dominant market: What we called a derivative market
in the past is now larger in sheer size than the primary market. The big question is what is
keeping the prices of the interest rate instruments tied to each other. That is, all of these
instruments are highly correlated. For instance, if the Federal Reserve drops the Fed funds
rate, then we may expect all of the short-term interest rates to fall. How do these rates move
together? The answer is no arbitrage, that is, the possibility does not exist for arbitrageurs
to take large positions in different securities whenever the prices across markets do not line
up. The concept of “line up” will become clear in future chapters. For now, we turn to
describing individual markets in more detail.
6
AN INTRODUCTION TO FIXED INCOME MARKETS
Figure 1.2
The Growth in Derivatives Markets: Notional
350
Swaps
300
Trillion of Dollars
250
OTC and
Exchange
Traded
Interest Rate
Options
200
150
Forward
Rate
Agreements
and Futures
100
50
0
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
Source: SIFMA and Bank for International Settlement
1.1.1 The Complexity of Fixed Income Markets
The previous section illustrates the growth in size of fixed income markets. The complexity
of fixed income markets is also extraordinary. Table 1.2 reports a snapshot of rates in the
U.S. fixed income markets on September 18, 2007. The table corresponds to screen
BTMM from Bloomberg terminals, and it is widely used by traders to quickly grasp the
relative positions of bond prices and interest rates across markets. The number of securities
described in this table is daunting. Starting from the top left corner, we have:
1. Federal funds rate quotes;
2. U.S. Treasury bill prices and yields at various maturities;
3. Eurodollar deposit rates at various maturities;
4. Repo and reverse repo rates;
5. U.S. Treasury bond yields and prices with various maturities;
6. Commercial paper quotes;
7. 90-day Eurodollar futures for various maturities;
8. Federal funds futures for various maturities;
9. LIBOR fixes;
10. Foreign exchange rates;
INTRODUCTION
7
11. 30-years mortgage backed securities;
12. 10-year Treasury note futures;
13. Swap rates for various maturities;
14. Other key rates, such as the prime.
How do all these market rates move together?
The notion is that these quantities are all highly correlated with the same events. For
instance, if there are worries of an increase in future inflation, we can expect the Federal
Reserve to increase the target Fed Funds rate (see Chapter 7). In turn, this expectation
as well as the rules of no arbitrage, discussed below, have an impact on other short-term
borrowing rates, such as the short-term LIBOR, the short-term Eurodollar rates and so on.
In this chapter we define only the terms appearing in Table 1.2. In the following chapters,
we describe the relations between these markets and many others that do not appear in
Table 1.2. The key concept is the concept of no arbitrage, which is helpful to introduce
right away.
1.1.2
No Arbitrage and the Law of One Price
At the source of the ripple-through effect from one market to the next is the notion of no
arbitrage. In its pure form, an arbitrage opportunity is defined as follows:
Definition 1.1 An arbitrage opportunity is a feasible trading strategy involving two or
more securities with either of the following characteristics:
1. It does not cost anything at initiation, and it generates a sure positive profit by a
certain date in the future;
2. It generates a positive profit at initiation, and it has a sure nonnegative payoff by a
certain date in the future.
The no arbitrage condition requires that no arbitrage opportunities exist.
A pure arbitrage trade consists in taking positions that generate, magically to some
extent, always nonnegative cash flows, and with certainty, some positive cash flow. There
are three key elements in Definition 1.1: The trade (1) costs nothing; (2) yields positive
profits with certainty; and (3) the profits arrive within a known time. For instance, if
an arbitrageur finds two securities that pay exactly $100 in six months, but one trades at
P1 = $97 and the other at P2 = $98, then an arbitrageur can apply the trader’s motto “buy
low and sell high,” and purchase 1 million units of Security 1 at $97 and sell 1 million units
of Security 2 at $98, realizing an inflow of $1 million. In six months, the two securities
generate exactly the same cash flow and therefore the trader is hedged: Whatever he or she
receives from Security 1 is then given to the holder of Security 2.
Of course, these types of pure arbitrage opportunities are hard to find in financial markets.
Because of transaction costs and the lack of perfect co-movement among variables, some
risks do in fact exist, and arbitrageurs must take them into account while they trade. The
rules of no arbitrage, however, are still key to defining some relationships that must exist
8
BID/ASK
LST/OPEN
HIGH/LOW
DJIA
FED FUNDS
5 1/8
5 1/8
5 1/8
13479.91
5 1/36
5 1/8
5 1/8
+76.49
US T-BILL YIELD/PRICE
4W
3.88
0.12
3.82
3M
4.16
0.02
4.07
6M
4.31
0.02
4.17
S&P 500 FUT
1947
+7.20
US BONDS YLD/BID/ASK/CHG
4 08/31/09
4.117
99-24+
99-25
4 1/2 05/15/10 4.128
100-29
100-29+
4 1/8 08/31/12 4.241
99-15
99-15+
4 3/4 08/15/17 4.493
102-00
102-01
5 05/15/37
4.735
104-05
104-06
CRB
324.31
SPOT FOREX
JPY
115.8000
EUR
1.3875
GBP
1.9983
CHF
1.1876
MXN
11.1030
CAD
1.0233
GNMN 6.0
GOLD 6.0
FNMA 6.0
Source: Bloomberg. Screen BTMM. Date: September 18, 2007.
A Snapshot of U.S. Treasury and Money Market Rates
-03
-04
-05+
-07
-19+
-.56
30Y MBS
100-24
100-25
100-09
100-10
100-07
100-08
DEALER CP
15D
5.060
30D
5.350
60D
5.400
90D
5.410
120D 5.380
180D 5.240
-02
00
-01
3.81
4.17
4.15
90D EUR $ FUT
DEC
95.07
MAR
95.38
JUN
95.51
SEP
95.56
DEC
95.56
MAR
95.51 ‘
10yr Note Fut
109 - 20
CRUDE OIL
NYM WTI 80.80
CBT
EURO $ DEP
3M
5.5000
6M
5.3300
1Y
5.0300
CCMP 2592.02
-04+
+.23
5.6000
5.4300
5.1300
+10.36
FUNDS FUT
SEP
95.01
OCT
95.11
NOV
95.30
DEC
95.39
JAN
95.44
FEB
95.56
SWAP RATES
3Y
4.809
5Y
4.918
10Y
5.174
REVERSE
O/N 5.15
1W
4.95
2W
4.85
1M
4.70
REPO
5.05
4.85
4.75
4.60
LIBOR FIX
1W
5.25875
1M
5.49625
2M
5.55375
3M
5.58750
4M
5.53625
5M
5.48313
6M
5.42000
1Y
5.11250
Key Rates
Prime
8.25
BLR
7.00
FDTR
5.25
Discount 5.75
AN INTRODUCTION TO FIXED INCOME MARKETS
Table 1.2
THE GOVERNMENT DEBT MARKETS
9
across assets prices, which in turn determine the relative prices of fixed income instruments.
In this book we will see how these rules of no arbitrage allow us to both compute the fair
value of fixed income instruments and to investigate their relative prices. Just as important,
we will focus on the impact that no arbitrage has on the risk of fixed income instruments
and therefore their risk management. At the basis of much of the analysis is the law of one
price, discussed next:
Fact 1.1 The law of one price establishes that securities with identical payoffs should
have the same price.
If the law of one price does not hold for some securities, then an arbitrage opportunity
exists. Indeed, the logic is the same as the one of the previous example: if two securities
have the same cash flows in the future but trade at different prices today, then an arbitrageur
could buy the underpriced security and sell the overpriced one, realizing a profit today.
Since the cash flows are the same in the future, the arbitrageur is perfectly hedged.
Before we investigate how no arbitrage and the law of one price allow us to study the
valuation, risk, and risk management practices of fixed income instruments, let’s take a
closer look at the fixed income markets, using the entries in Table 1.2 as a guide. We begin
with government debt, appearing under the heading of U.S. T-Bills and U.S. Bonds in Table
1.2.
1.2
THE GOVERNMENT DEBT MARKETS
Essentially all countries issue debt to finance their operations. U.S. government debt has
always occupied a special place in fixed income markets, mainly because it is perceived to
have an extremely low probability of default. That is, investing in U.S. Treasury securities
is considered “safe,” as the government will repay its debt to investor. The quotation marks
around the word “safe” underly an important caveat, though, which is what makes the
analysis of fixed income securities so interesting. A U.S. Treasury bond can be considered
a “safe” investment in terms of its risk of default: As noted, the issuer will in all likelihood
repay its debt to its creditors (investors). The rationale behind its default safety is that these
bonds are backed by the ability of the U.S. government to levy taxes on its citizens in the
future to pay the debt back.
However, an investment in a U.S. Treasury bond may not be safe in terms of its return
on investment over a short period of time. To provide an example, Figure 1.3 plots the life
cycle of a 20-year bond, from its issuance in February 1986 to its maturity on February
2006.1 The variation over time of the price of the bond is quite stunning, with run ups of
over 30 percent within two years (e.g., between 1991 and 1993), and run downs at even
higher speed (e.g., 1994). An investor buying this bond in 1993 would have suffered severe
capital losses within the next year or so.
In addition to the potential capital losses in the bond price during a shorter period of
time than the bond’s maturity, an investment in U.S. Treasury securities entails additional
sources of risk. The first is that most of the Treasury securities are nominal securities,
that is, they pay coupons and principal in dollars. Therefore, if between the purchase of
1 Data
excerpted from CRSP (Daily Treasuries) ¤2009 Center for Research in Security Prices (CRSP), The
University of Chicago Booth School of Business.
10
AN INTRODUCTION TO FIXED INCOME MARKETS
Figure 1.3
The February 2006, 9.375%, 20-Year Bond Price Path
135
130
125
Price
120
115
110
105
100
95
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
Source: Center for Research in Security Prices
the bond and its maturity (e.g., 30 years) the U.S. enters a period of sustained inflation,
the effective value of coupons and principal decreases, as investors cannot purchase as
many consumption goods. This inflation risk must be taken into account in the analysis of
Treasury securities. A related risk concerns the fact that the coupon and principal payments
are in U.S. dollars, which may entail a currency risk for an overseas investor. For instance,
European investors who purchased safe U.S. Treasury notes in 2005 have been hurt by the
devaluation of the dollar with respect to the euro between 2006 and 2008.
The U.S. government issues various types of securities. Table 1.3 lists the types of
securities. Treasury bills (T-bills) are short-term debt instruments, with maturity up to one
year.2 They do not pay any cash flow over time, only the principal at maturity. Treasury
bills are issued very frequently, typically every week for bills up to six months, and every
four weeks for one-year bills.
Treasury notes (T-notes) are medium-long term debt instruments, with maturity up to 10
years. These notes carry a fixed coupon that is paid semi-annually up to the maturity of the
note. They are issued every month, except the ten-year note, that is issued less frequently.
Treasury bonds (T-bonds) are longer-term debt instruments with maturity of 30 years at
issuance. As with Treasury notes, bonds also carry a semi-annual coupon. The Treasury
issues these long-term bonds every six months.
In 1997 the U.S. government started issuing TIPS – Treasury Inflation Protected Securities – that is, securities that are indexed to inflation. Investors in T-notes and T-bonds are
subject to inflation risk: Because the payment of coupons and final principal is in nominal
2 The
Treasury suspended the issuance of one year Treasury bills from August 2001 to June 2008.
THE GOVERNMENT DEBT MARKETS
11
Table 1.3 U.S. Treasury Debt Securities
Name
Maturity
Coupon Rate
Principal
Treasury Bills
Treasury Notes
Treasury Bonds
TIPS
4, 13, 26, and 52 weeks
2, 5, and 10 years
30 years
5, 10, and 20 years
None
Fixed, semi-annual
Fixed, semi-annual
Fixed, semi-annual
Fixed
Fixed
Fixed
Adjusted for inflation
terms (i.e., simply U.S. dollars), if inflation increases substantially during the life of the
debt instruments, these sums of money will be able to buy less of consumer goods. The
TIPS offer protection to investors against this possibility: Because the principal is adjusted
for inflation, higher inflation translates into both a higher final payoff at maturity of the
bond, and higher coupons as well, since the coupon is defined as a fixed percentage of
current principal (which increases with inflation). TIPS are issued with maturities of 5, 10
and 20 years.
The issuance calendar of the U.S. Treasury is very dense. Table 1.4 provides a snapshop
of the issuing activity of the U.S. Treasury as of July 15, 2009. Market participants refer to
the most recently issued Treasury securities as on-the-run securities, while all the others
are called off-the-run. On-the-run Treasury securities tend to trade at a premium compared
to similar off-the-run Treasury securities, which tend to be less liquid than the on-the-run
securities.
1.2.1
Zero Coupon Bonds
Zero coupon bonds are securities that pay only the principal at maturity. A simple example
is the Treasury bill described in Table 1.3. Other zero coupon bonds are available in the U.S.
market through the STRIPS program. STRIPS (Separate Trading of Registered Interest and
Principal Securities) are zero coupon bonds created from available U.S. Treasury notes and
bonds by splitting the principal and each of the coupons from the bond. The U.S. Treasury
does not issue these securities directly to investors, but investors can purchase them and
hold them through financial institutions and government securities brokers and dealers. As
an example of the available STRIPS on a particular date, Table 1.5 reports the stripped
coupons available on September 25, 2008. The stripped coupon are the zero coupon bonds
that are created only from the coupon interest payments of Treasury notes and bonds. In
addition, a similar table is available for the stripped principals. The availability of these
zero coupon bond securities with maturity up to 30 years enables investors to be more
effective in their investment strategies and in their risk management practices, as we will
discuss in later chapters.
1.2.2
Floating Rate Coupon Bonds
The bonds issued by the U.S. government have a fixed coupon rate. A floating rate coupon
bond is like a standard coupon bond, but its coupon is indexed to some other short-term
interest rate, which changes over time. While the U.S. government does not issue floating
rate bonds, other governments do. For instance, Italy issues the CCT bond, which is an
Italian Treasury debt security whose coupon rate is indexed to the six month rate of Italian 6-
12
AN INTRODUCTION TO FIXED INCOME MARKETS
Table 1.4
Security
3-YEAR
9-YEAR
10-YEAR
29-YEAR
2-YEAR
5-YEAR
7-YEAR
3-YEAR
9-YEAR
29-YEAR
2-YEAR
5-YEAR
7-YEAR
3-YEAR
10-YEAR
30-YEAR
2-YEAR
5-YEAR
5-YEAR
7-YEAR
3-YEAR
9-YEAR
9-YEAR
2-YEAR
5-YEAR
7-YEAR
3-YEAR
9-YEAR
29-YEAR
2-YEAR
5-YEAR
7-YEAR
3-YEAR
10-YEAR
30-YEAR
2-YEAR
5-YEAR
20-YEAR
3-YEAR
9-YEAR
Issuance Acitivity of Bonds, Notes, and TIPS: January 15, 2009 to July 15, 2009
Term
Type
NOTE
10-MONTH NOTE
TIPS
10-MONTH BOND
NOTE
NOTE
NOTE
NOTE
11-MONTH NOTE
11-MONTH BOND
NOTE
NOTE
NOTE
NOTE
NOTE
BOND
NOTE
TIPS
NOTE
NOTE
NOTE
9-MONTH TIPS
10-MONTH NOTE
NOTE
NOTE
NOTE
NOTE
11-MONTH NOTE
11-MONTH BOND
NOTE
NOTE
NOTE
NOTE
NOTE
BOND
NOTE
NOTE
TIPS
NOTE
10-MONTH NOTE
Issue
Date
Maturity
Date
7/15/2009
7/15/2009
7/15/2009
7/15/2009
6/30/2009
6/30/2009
6/30/2009
6/15/2009
6/15/2009
6/15/2009
6/1/2009
6/1/2009
6/1/2009
5/15/2009
5/15/2009
5/15/2009
4/30/2009
4/30/2009
4/30/2009
4/30/2009
4/15/2009
4/15/2009
4/15/2009
3/31/2009
3/31/2009
3/31/2009
3/16/2009
3/16/2009
3/16/2009
3/2/2009
3/2/2009
3/2/2009
2/17/2009
2/17/2009
2/17/2009
2/2/2009
2/2/2009
1/30/2009
1/15/2009
1/15/2009
7/15/2012
5/15/2019
7/15/2019
5/15/2039
6/30/2011
6/30/2014
6/30/2016
6/15/2012
5/15/2019
5/15/2039
5/31/2011
5/31/2014
5/31/2016
5/15/2012
5/15/2019
5/15/2039
4/30/2011
4/15/2014
4/30/2014
4/30/2016
4/15/2012
1/15/2019
2/15/2019
3/31/2011
3/31/2014
3/31/2016
3/15/2012
2/15/2019
2/15/2039
2/28/2011
2/28/2014
2/29/2016
2/15/2012
2/15/2019
2/15/2039
1/31/2011
1/31/2014
1/15/2029
1/15/2012
11/15/2018
Interest Yield
Rate % %
1.5
3.125
1.875
4.25
1.125
2.625
3.25
1.875
3.125
4.25
0.875
2.25
3.25
1.375
3.125
4.25
0.875
1.25
1.875
2.625
1.375
2.125
2.75
0.875
1.75
2.375
1.375
2.75
3.5
0.875
1.875
2.625
1.375
2.75
3.5
0.875
1.75
2.5
1.125
3.75
1.519
3.365
1.92
4.303
1.151
2.7
3.329
1.96
3.99
4.72
0.94
2.31
3.3
1.473
3.19
4.288
0.949
1.278
1.94
2.63
1.385
1.589
2.95
0.949
1.849
2.384
1.489
3.043
3.64
0.961
1.985
2.748
1.419
2.818
3.54
0.925
1.82
2.5
1.2
2.419
Price
per $100
CUSIP
99.944485
97.998772
99.592335
99.104142
99.94874
99.651404
99.510316
99.753523
92.968581
92.50169
99.871675
99.718283
99.689717
99.713432
99.44721
99.36198
99.853739
100.113235
99.691687
99.968223
99.970714
103.325496
98.298568
99.853739
99.529266
99.942292
99.667005
97.504473
97.456658
99.830481
99.479306
99.22194
99.871395
99.411068
99.264139
99.901394
99.667162
99.063837
99.77965
111.579767
912828LB4
912828KQ2
912828LA6
912810QB7
912828LF5
912828KY5
912828KZ2
912828KX7
912828KQ2
912810QB7
912828KU3
912828KV1
912828KW9
912828KP4
912828KQ2
912810QB7
912828KL3
912828KM1
912828KN9
912828KR0
912828KK5
912828JX9
912828KD1
912828KH2
912828KJ8
912828KT6
912828KG4
912828KD1
912810QA9
912828KE9
912828KF6
912828KS8
912828KC3
912828KD1
912810QA9
912828JY7
912828JZ4
912810PZ5
912828KB5
912828JR2
Source: U.S.Treasury Web Site http://www.treasurydirect.gov/RI/OFNtebnd accessed on July 16, 2009.
THE GOVERNMENT DEBT MARKETS
Table 1.5
13
Stripped Coupon Interest on September 25, 2008
Maturity
Year Month Day Bid
Ask
2008
11
15 99.898 99.918
2009
2
15 99.478 99.498
2009
5
15 98.979 98.999
2009
8
15 98.473 98.493
2009
11
15 97.982 98.002
2010
2
15 97.487 97.507
2010
5
15 96.879 96.899
2010
8
15 96.294 96.314
2010
11
15 95.722 95.742
2011
2
15 94.83 94.85
2011
5
15 94.304 94.324
2011
8
15 93.274 93.294
2011
11
15 92.957 92.977
2012
2
15 91.072 91.092
2012
5
15 90.705 90.725
2012
8
15 89.274 89.294
2012
11
15 88.498 88.518
2013
2
15 87.478 87.498
2013
5
15 86.684 86.704
2013
8
15 85.988 86.008
2013
11
15 85.014 85.034
2014
2
15 83.999 84.019
2014
5
15 83.172 83.192
2014
8
15 82.185 82.205
2014
11
15 81.257 81.277
2015
2
15 79.706 79.726
2015
5
15 78.898 78.918
2015
8
15 77.972 77.992
2015
11
15 76.772 76.792
2016
2
15 75.885 75.905
2016
5
15 74.437 74.457
2016
8
15 73.593 73.613
2016
11
15 72.086 72.106
2017
2
15 71.16 71.18
2017
5
15 70.144 70.164
2017
8
15 69.036 69.056
2017
11
15 68.213 68.233
2018
2
15 67.643 67.663
2018
5
15 66.816 66.836
2018
8
15 65.674 65.694
2018
11
15 64.851 64.871
2019
2
15 63.626 63.646
2019
5
15 62.904 62.924
2019
8
15 61.826 61.846
2019
11
15 61.081 61.101
2020
2
15 60.194 60.214
2020
5
15 59.29 59.31
2020
8
15 58.475 58.495
2020
11
15 57.716 57.736
2021
2
15 56.876 56.896
2021
5
15 56.128 56.148
2021
8
15 55.368 55.388
2021
11
15 54.649 54.669
2022
2
15 53.989 54.009
2022
5
15 53.282 53.302
2022
8
15 52.599 52.619
2022
11
15 51.869 51.889
2023
2
15 51.144 51.164
2023
5
15 50.606 50.626
Source: The Wall Street Journal.
Chg Asked Yield
0.001
0.6
-0.068
1.31
-0.056
1.59
-0.146
1.72
-0.194
1.78
-0.236
1.83
-0.277
1.93
-0.318
2
-0.359
2.05
-0.413
2.23
-0.442
2.23
-0.539
2.42
-0.481
2.34
-0.48
2.78
-0.515
2.69
-0.566
2.94
-0.589
2.97
-0.607
3.07
-0.647
3.1
-0.666
3.11
-0.725
3.18
-0.763
3.26
-0.814
3.29
-0.828
3.36
-0.903
3.41
-0.462
3.58
-0.489
3.6
-0.502
3.64
-0.525
3.73
-0.538
3.77
-0.573
3.9
-0.599
3.92
-0.707
4.06
-0.483
4.09
-0.491
4.14
-0.482
4.21
-0.505
4.23
-0.468
4.21
-0.474
4.23
-0.559
4.3
-0.565
4.32
-0.601
4.4
-0.609
4.4
-0.596
4.46
-0.602
4.47
-0.312
4.51
-0.314
4.54
-0.316
4.56
-0.319
4.58
-0.373
4.61
-0.393
4.62
-0.396
4.64
-0.416
4.65
-0.454
4.66
-0.475
4.67
-0.477
4.68
-0.479
4.7
-0.481
4.71
-0.484
4.7
Maturity
Year Month Day Bid
Ask
Chg Asked Yield
2023
8
15 50.039 50.059 -0.487
4.7
2023
11
15 49.424 49.444 -0.489
4.71
2024
2
15 48.815 48.835 -0.529
4.71
2024
5
15 48.286 48.306 -0.532
4.71
2024
8
15 47.746 47.766 -0.553
4.7
2024
11
15 47.194 47.214 -0.555
4.7
2025
2
15 46.797 46.817 -0.408
4.69
2025
5
15 46.221 46.241 -0.371
4.69
2025
8
15 45.537 45.557 -0.372
4.71
2025
11
15 44.972 44.992 -0.297
4.72
2026
2
15 44.357 44.377 -0.297
4.73
2026
5
15 43.879 43.899 -0.298
4.72
2026
8
15 43.332 43.352 -0.298
4.73
2026
11
15 42.828 42.848 -0.299
4.73
2027
2
15 42.445 42.465 -0.262
4.71
2027
5
15 41.934 41.954 -0.263
4.72
2027
8
15 41.467 41.487 -0.263
4.71
2027
11
15 41.025 41.045 -0.206
4.71
2028
2
15 40.685 40.705 -0.11
4.69
2028
5
15 40.216 40.236 -0.11
4.69
2028
8
15 39.694 39.714 -0.111
4.7
2028
11
15 39.178 39.198 -0.111
4.71
2029
2
15 38.686 38.706 -0.092
4.71
2029
5
15 38.277 38.297 -0.033
4.71
2029
8
15 37.815 37.835 -0.033
4.71
2029
11
15 37.493 37.513 -0.015
4.69
2030
2
15 37.138 37.158 -0.015
4.68
2030
5
15 36.749 36.769 -0.015
4.68
2030
8
15 36.463 36.483 0.063
4.66
2030
11
15 36.084 36.104 0.063
4.66
2031
2
15 35.691 35.711 0.063
4.65
2031
5
15 35.282 35.302 -0.074
4.65
2031
8
15 34.957 34.977 -0.074
4.64
2031
11
15 34.656 34.676 -0.074
4.63
2032
2
15 34.438 34.458 -0.074
4.61
2032
5
15 34.264 34.284 -0.075
4.58
2032
8
15 33.918 33.938 -0.075
4.58
2032
11
15 33.556 33.576 -0.075
4.57
2033
2
15 33.337 33.357 -0.076
4.55
2033
5
15 32.984 33.004 -0.075
4.55
2033
8
15 32.674 32.694 -0.076
4.54
2033
11
15 32.329 32.349 -0.076
4.54
2034
2
15 32.008 32.028 -0.075
4.54
2034
5
15 31.651 31.671 -0.075
4.54
2034
8
15 31.357 31.377 -0.076
4.53
2034
11
15 31.008 31.028 -0.075
4.53
2035
2
15 30.662 30.682 -0.076
4.53
2035
5
15 30.321 30.341 -0.075
4.53
2035
8
15 29.983 30.003 -0.075
4.53
2035
11
15 29.649 29.669 -0.075
4.53
2036
2
15 29.339 29.359 -0.075
4.53
2036
5
15 29.268 29.288 0.161
4.49
2036
8
15 28.865 28.885 0.389
4.5
2036
11
15 28.664 28.684 0.161
4.49
2037
2
15 28.249 28.269 0.384
4.5
2037
5
15 28.212 28.232 0.161
4.47
2037
8
15 27.901 27.921 0.386
4.47
2038
2
15 27.331 27.351 0.23
4.46
14
AN INTRODUCTION TO FIXED INCOME MARKETS
month T-bills (BOT). In addition, interest rate floaters are issued by financial institutions and
corporations, as well as government agencies, such as the government mortgage companies
Ginnie Mae, Freddie Mac, and Fannie Mae, within their collateralized mortgage obligations
programs.
1.2.3 The Municipal Debt Market
The U.S. federal government issues debt to finance federal government expenses, such as
health care and military expenses. Individual municipalities also issue debt independently
to finance local projects. For instance, the City of Chicago issued bonds for $983,310,000
in 2003 to pay for an expansion project of its O’Hare International Airport.
The most interesting feature of “muni” bonds is that the interest income from their
coupons is tax-exempt. As a consequence, the yield is lower than other regular Treasury
notes and bonds, as the latter pay an income that is taxable according to investors’ income
tax rates.
1.3 THE MONEY MARKET
When we speak of the money market, we refer to the market for short-term borrowing and
lending. Banks and financial institutions have various means of borrowing and lending at
any point in time. The entries in Table 1.2 summarize these channels.
1.3.1 Federal Funds Rate
Banks and other financial institutions must keep some amount of capital within the Federal
Reserve. Balances at the Federal Reserve yield a small rate of return, which was in fact
zero until September 2008. It is in the interest of banks to maintain their reserves as close to
the limit as possible. Banks with a reserve surplus may then lend some of their reserves to
banks with a reserve deficit. The effective Federal funds rate is the size-weighted average
rate of interest that banks charge to each other to lend or borrow reserves at the Federal
Reserve. Chapter 7 describes this market in more detail.
1.3.2 Eurodollar Rate
The Eurodollar rate is the rate of interest on a dollar deposit in a European-based bank.
These are short-term deposits, ranging from 3 months to one year. In particular, the 90-day
Eurodollar rate has become a standard reference to gauge the conditions of the interbank
market. For instance, the market for Eurodollar futures and options, financial derivatives
traded at the Chicago Mercantile Exchange that allow financial institutions to bet on or
hedge against the future evolution of the Eurodollar rate (see Chapter 6), is among the
largest and most liquid derivative markets in the world.
1.3.3 LIBOR
LIBOR stands for London Interbank Offer Rate. The British Bankers Association publishes
daily the LIBOR rates. These rates correspond to the average interest rate that banks charge
to each other for short-term uncollateralized borrowing in the London market. The rates
THE REPO MARKET
15
available are very similar to the Eurodollar rates (see Table 1.2). LIBOR is however one
of the most important benchmark rates, used often as the reference index in the large
over-the-counter derivatives market. As explained in Chapter 5, interest rate swaps, the
single largest derivatives market (see Table 1.1), use LIBOR rates as the reference rates to
determine the size of cash flows implied by a contract.
1.4
THE REPO MARKET
The last entry in the top row of Table 1.2 reports the repo and reverse repo rates. The
repo market plays an important role in the fixed income industry, as it is used by traders
to borrow and lend cash on a collateralized basis. Because borrowing is collateralized, it
is considered a safer way to lend cash, and this contributed to its growth over the years,
making the repo market one of the most important sources of financing for traders. First of
all, the formal definition:
Definition 1.2 A repurchase agreement (repo) is an agreement to sell some securities to
another party and buy them back at a fixed date and for a fixed amount. The price at which
the security is bought back is greater than the selling price and the difference implies an
interest rate called the repo rate.
A reverse repo is the opposite transaction, namely, it is the purchase of the security
for cash with the agreement to sell it back to the original owner at a predetermined price,
determined, once again, by the repo rate.
The best way to understand a repo transaction is to consider it as collateralized borrowing.
A trader entering into a repo transaction with a repo dealer is borrowing cash (the sale price)
in exchange for the security, which is held hostage by the repo dealer. If at the end of the
repo term the trader were to default, the repo dealer could sell the security and be made
whole. The following example illustrates the trade:
EXAMPLE 1.1
Suppose that a trader on September 18, 2007 (time t) wants to take a long position
until a later time T on a given U.S. security, such as the 30-year Treasury bond. Let
Pt denote the (invoice) price of the bond at time t. Figure 1.4 provides a schematic
representation of the repo transaction: At time t, the trader buys the bond at market
price Pt and enters a repurchase agreement with the repo dealer. Hence, the trader
delivers the bond as collateral to the repo dealer and receives the cash to purchase the
bond. In fact, the repo dealer typically gives something less than the market price
of the bond, the difference being called a haircut. At time t the trader and the repo
dealer agree that the trader will return back the amount borrowed, (Pt − haircut),
plus the repo rate.
What happens then at time T ? At time T , the trader gets back the bond from the
repo dealer, sells the bond in the market to get PT and pays (Pt − haircut) plus the
repo interest to the dealer. The repo interest is computed as the repo rate agreed at
time t times the time between t and T . For instance, if n days pass between the two
16
AN INTRODUCTION TO FIXED INCOME MARKETS
Figure 1.4
Schematic Repo Transaction
time t
MARKET
buy bond at Pt
=⇒
⇐=
pay Pt
TRADER
sell bond at PT
⇐=
=⇒
get PT
get the bond
⇐=
TRADER
REPO DEALER
=⇒
pay (Pt − haircut)× (1+repo rate × 3n6 0 )
deliver bond
=⇒
⇐=
get Pt − haircut
REPO DEALER
time T = t + n days
MARKET
dates, we have
Repo interest =
n
× Repo rate × (Pt − haircut)
360
(1.1)
where the denumerator “360” stems from the day count convention in the repo market.
The profit to the trader is then PT − Pt −Repo interest. In percentage terms, the
trader only put up the haircut (the margin) as own capital. Hence, the return on capital
is
PT − Pt − Repo interest
Return on capital for trader =
Haircut
The position is highly leveraged and entails quite large risks. The case study in
Section 3.7 of Chapter 3 discusses the risk and return of such leveraged transactions.
The term T of the repo transaction is decided at initiation, i.e., time t. In particular, most
repurchase agreements are for a very short term, mainly overnight. However, as shown in
Table 1.2, longer-term agreements reach 30 days or even more. Recall also that the repo
rate is decided at time t.
Between t and T the trader (who is long the bond) earns the interest that accrues on
the bond. Because the trader has to pay the repo rate during this period, setting up the
repo transaction tends to generate a positive or negative stream of payments, depending on
whether the interest earned on the bond is above or below the repo interest. We say that the
trade implies a positive carry if the interest on the bond is above the repo rate and negative
carry if the interest on the bond is below the repo rate.
1.4.1 General Collateral Rate and Special Repos
Other important definitions and characteristics of repo markets are as follows:
1. General Collateral Rate (GCR): This is the repo rate on most Treasury securities,
such as the off-the-run Treasuries. Because most Treasury securities have similar
THE REPO MARKET
17
characteristics in terms of liquidity, market participants require the same interest rate
for collateralized borrowing.
2. Special Repo Rate: At times, one particular Treasury security is in high demand
and hence the repo rate on that security falls to a level substantially below the GCR.
As an example, on-the-run Treasury securities typically are “on special,” in the sense
that the repo rate charged for collateralized borrowing is smaller than the GCR.
Why does a security that is in high demand entail a lower (special) repo rate? To
understand the logic, consider the next example, which entails a reverse repo, whose rates
are also quoted in Table 1.2.
EXAMPLE 1.2
Consider a trader who thinks a particular bond, such as the on-the-run 30-year
Treasury bond, is overpriced and wants to take a bet that its price will decline in the
future. If the trader does not have the bond to sell outright, then he or she can enter
into a reverse repo with a repo dealer to obtain the bond to sell. More specifically,
in a reverse repo, the trader essentially (A) borrows the security from the dealer; (B)
sells it in the market; and (C) posts cash collateral with the dealer. Figure 1.5 shows
a schematic representation of the trade.
The trader is now lending money to the repo dealer against the bond. Therefore,
the trader is now entitled to receive the repo rate. However, the trader, who wants to
speculate on the decrease in the bond price, is happy to forgo part or all of the repo
rate in order to get hold of the bond. If many traders want to undertake the same
strategy of shorting that particular bond, then that bond is in high demand, and the
repo rate for that bond declines below the general collateral rate. That bond is said
to be “on speacial.” The profit from the reverse repo transaction is then
Profit = (Pt − PT ) + Repo interest
where the repo interest is computed as in Equation 1.1, namely, the amount deposited
with the repo dealer (Pt ) times the repo rate times n/360, where n is the number of
days between the two trading dates t and T .
As mentioned, the repo market has grown steadily over the years. Table 1.6 shows the
average daily amount outstanding in these contracts.3 Since borrowing is collateralized by
the value of the asset, the repo rate is lower than other borrowing rates available to banks,
such as LIBOR. Figure 1.6 plots the time series of the one month and the three month
T-Bill, Repo and LIBOR rates from May 1991 to April 2008. As it can be seen, for both
maturities, the safe T-bill rate is the smallest and the LIBOR is the highest of the three rates,
as borrowing and lending at the LIBOR rate are riskier as the loans are uncollateralized.
We cover additional details regarding the repo market and its uses by market participants
in a number of case studies. For instance, in Chapters 3 and 4 we discuss the use of
repurchase agreements to increase portfolio leverage, in Chapter 5 we illustrate the use
of the repo market to carry out a swap spread arbitrage trade, and in Chapter 16 we use
3 The
amount outstanding of repurchase agreements need not equal the amount outstanding of reverse repurchase
agreements, as each column reflects the size of collateralized borrowing or lending of security dealers only, and
not the whole universe of repo counterparties.
18
AN INTRODUCTION TO FIXED INCOME MARKETS
Figure 1.5
Reverse Repo Transaction
time t
MARKET
sell bond at Pt
⇐=
=⇒
get Pt
TRADER
buy bond at PT
=⇒
⇐=
pay PT
give bond back
=⇒
TRADER
REPO DEALER
⇐=
get back Pt ×(1+ repo rate × 3n6 0 )
borrow bond
⇐=
REPO DEALER
=⇒
use Pt as cash collateral
time T = t + n days
MARKET
repo transactions to implement a relative value arbitrage trade on the yield curve through a
dynamic long/short strategy.
1.4.2 What if the T-bond Is Not Delivered?
Consider the reverse repurchase agreement displayed in Figure 1.5. At maturity of the repo
contract (time T ), the trader must return the bond to the repo dealer in exchange for the
n
× repo rate). What happens if the trader does not return the
cash amount Pt × (1 + 360
security? Such an occurrence is called a fail, and up to May, 2009 such a failure to deliver
would have simply implied that the repo dealer in this example would have kept the cash
received, Pt , plus the repo interest. The cost for failing to deliver for the trader was simply
to forgo the repo interest. The financial crisis of 2007 - 2009 led the Federal Reserve to
lower the reference Fed funds rate to close to zero, and repo rates also fell to essentially
zero. When the repo rate is zero, however, the cost for a trader to fail to deliver the bond
is very small, as the trader may keep the bond itself if this bond is particularly valuable.
The financial crisis of 2007 - 2009 generated a ‘flight-to-quality,’ meaning that investors
dumped all risky securities and strongly demanded safe U.S. Treasuries, as the demand for
safe collateral increased. This increase in demand for U.S. Treasuries made it difficult or
costly for traders who have short positions to find the bonds to return to their counterparties
in the reverse repo transactions. Given the small cost of failing to deliver, the number of
fails spiked in the last quarter of 2008. Figure 1.7 reports the cumulative weekly failures
of delivering Treasury securities, in millions of dollars, by primary dealers, and the 2008
spike is clearly visible. The figure however also shows that sustained periods of delivery
fails occurred in the past as well, such as in 2001 and in 2003. We should mention that a
spike in delivery fails can also be due to a snowball effect, as the failure to deliver from
a security dealer implies that another security dealer who was counting on the delivery to
THE REPO MARKET
Table 1.6 Financing by U.S. Government Securities Dealers
Reverse Repurchase and Repurchase Agreements (1)
Average Daily Amount Outstanding
1981 - 2006 ($ Billions)
Reverse Repurchase
Repurchase
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
46.7
75.1
81.7
112.4
147.9
207.7
275
313.6
383.2
377.1
417
511.1
594.1
651.2
618.8
718.1
883
1,111.4
1,070.1
1,093.3
1,311.3
1,615.7
1,685.4
2,078.5
2,355.2
2,225.2
65.4
95.2
102.4
132.6
172.9
244.5
292
309.7
398.2
413.5
496.6
628.2
765.6
825.9
821.5
973.7
1,159.0
1,414.0
1,361.0
1,439.6
1,786.5
2,172.4
2,355.7
2,868.2
3,288.4
3,388.3
(1) Figures cover financing involving U.S. government, federal
agency, and federal agency MBS securities.
Source: Federal Reserve Bank of New York
obtained from SIFMA web site: http://www.sifma.net/story.asp?id=1176
Total
112.1
170.3
184.1
245
320.8
452.2
567
623.3
781.4
790.5
913.6
1,139.3
1,359.7
1,477.1
1,440.3
1,691.8
2,042.0
2,525.5
2,431.1
2,532.9
3,097.7
3,788.1
4,041.1
4,946.7
5,643.6
5,613.5
19
20
AN INTRODUCTION TO FIXED INCOME MARKETS
Figure 1.6 Short-Term Rates: 1991 - 2008
Panel A. One Month Rates
Interest Rate (%)
8
6
4
2
0
Treasury
Repo
LIBOR
1992
1994
1996
1998
2000
2002
2004
2006
2008
2004
2006
2008
Panel B. Three Month Rates
Interest Rate (%)
8
6
4
2
0
Treasury
Repo
LIBOR
1992
1994
1996
1998
2000
2002
Source: Federal Reserve Board, British Bankers Association, Bloomberg.
THE MORTGAGE BACKED SECURITIES MARKET AND ASSET-BACKED SECURITIES MARKET
Figure 1.7
21
Primary Dealers Fails to Deliver: 1990 to 2009
6
3
x 10
Fails (in Millions of Dollars)
2.5
2
1.5
1
0.5
0
07/04/1990
07/07/1993
07/03/1996
07/07/1999
07/03/2002
07/06/2005
07/02/2008
Source: Federal Reserve Bank of New York.
settle his or her own obligation may be unable to deliver as well, and so on, generating a
domino effect.4
Starting May 1, 2009, the Federal Reserve imposed a penalty charge of 3% over its
Fed funds rate for failing to deliver the bonds in the repo transactions the Federal Reserve
conducts daily in its open market operations (see Chapter 7 for details on the Fed conduct
of monetary policy). In addition, the Federal Reserve has been encouraging market participants to adopt a similar charge as part of best practices in repo market transactions. An
interesting outcome of such a penalty on a failure to deliver is that on May 1, 2009, the repo
rate for some Treasury securities that were on special became negative.5 How is it possible
that an interest rate is negative? The reason is that a trader who has to deliver a given Treasury security to a conterparty is willing to pay to get hold of the security rather than incurring
the penalty. In particular, the trader can enter into a reverse repo with another repo dealer
at a negative repo rate to obtain the Treasury security to deliver to the original counterparty.
1.5
THE MORTGAGE BACKED SECURITIES MARKET AND
ASSET-BACKED SECURITIES MARKET
One of the interesting patterns evident in Figure 1.1 is the dramatic growth experienced by
the mortgage backed securities market, which hit the $8.9 trillion mark by the end of 2008.
Chapter 8 describes this market in detail as well as the type of securities that are exchanged
4 See the “Guide to FR2004 Settlement Fails Data,”
Federal Reserve of New York. See also the article by Michael
J. Fleming and Kenneth D. Garbade, “When the Back Office Moved to the Front Burner: Settlement Fails in the
Treasury Market after 9/11,” Federal Reserve Bank of New York Economic Policy Review, November 2002.
5 See Bloomberg.com Web site http://www.bloomberg.com/apps/news?pid=20601009&sid=a85sg4IKcjCM.
22
AN INTRODUCTION TO FIXED INCOME MARKETS
in it. However, as a brief introduction, the source of the mortgage-backed securities market
is relatively simple: homeowners across the U.S. finance their homes through mortgages,
issued by local savings & loans, thrifts, and other banks. When a bank issues a mortgage
to a homeowner, the mortgage rests on the asset side of the bank’s balance sheet. The
mortgage is a fixed income instrument: It is a promise from the homeowner to make
certain cash payments in the future. These cash payments are affected by numerous events,
discussed further in Chapter 8, which make them risky for the bank. In particular, if a
local bank provides mortgages to a local community only, it is subject to the risk that these
homeowners may all default at the same time because of local geographical factors. For
instance, if the local community is highly specialized in a particular industrial sector, and
the latter goes into an economic crisis, one could expect large layoffs in that community,
which in turn would increase the probability that homeowners will default on their mortgage
payment obligations. Similarly, if the house prices of that particular community decline,
the collateral in the mortgage contract declines, and the local bank is then in a more risky
position than before.
Mortgage backed securities allow a bank to diversify this risk. The idea is to resell its
mortgages, now on the asset side of the bank, for cash. In order to improve the liquidity
and to mitigate credit risk, the market evolved into one in which many similar mortgages
are pooled together to form a large collateral of assets. These assets, which have better
characteristics in terms of diversification of risk, make up the collateral on debt securities
issued to individual investors, called mortgage backed securities. In summary, an investor
in a mortgage backed security obtains a legal claim to the cash flows (coupons) that are
paid by the original homeowner.
The mortgage backed securities trade in the market. For instance, in Table 1.2, the
heading “30 Y MBS” reports the prices of popular mortgage-backed securities, those
issued by Ginnie Mae (GNMN 6.0), Freddie Mac (GOLD 6.0) and Fannie Mae (FNMA
6), the three largest players in the mortgage-backed securities market. From an investment
perspective, a large part of the mortgage-backed securities market is considered default
free, because these three big players – Ginnie, Freddie and Fannie – have an implicit or
explicit backing of the full faith of the U.S. government. Indeed, while Ginnie Mae has
always been a government entity, Fannie and Freddie entered conservatorship in September
2008, which implies that their own debt securities but especially their mortgage-backed
securities are default free. Still, compared to Treasury debt securities mortgage-backed
securities have many peculiarities regarding the timing of promised cash flows, which
may vary unexpectedly due to changes in interest rates, or changes in housing prices,
or a severe recession. These unexpected variations in cash flows make mortgage-backed
securities risky and, for this reasons, such securities typically offer an additional return
on investment, compared to Treasury securities. Chapter 8 discusses this market in more
detail.
Similarly to the mortgage backed securities market, the asset backed securities market
involves the issuance of debt instruments to investors, collateralized by various types of
loans, such as auto loans, credit cards, and the like. The market is smaller in size, as shown
in Table 1.1.
THE DERIVATIVES MARKET
23
Table 1.7 Borrowing rates for firm A and B
Fixed Rate
Floating Rate
1.6
Firm A
Firm B
15%
LIBOR + 3%
12%
LIBOR + 2%
THE DERIVATIVES MARKET
Table 1.2 also reports quotes of several derivative securities. As show in Table 1.1, the
interest rate derivatives market is huge and it has been growing steadily for the past three
decades, as illustrated in Figure 1.2. Chapters 5 and 6 explore these markets in more
detail, and provide some early examples of uses of derivative contracts for corporations
and traders. Additional examples discussing the fair valuation and the risk of derivative
securities are offered throughout the book, as we explore the modeling devices applied by
financial institutions to price and hedge these securities.
The swap contract is the largest market of all. While future chapters discuss the pricing,
hedging and the risk involved in swaps, it is informative at this point to see the economic
need that led to the creation of this market at the beginning of the 1980s.
1.6.1
Swaps
Interest rate swap contracts were introduced in the early 1980s to take advantage of some
apparent arbitrage opportunity that was surfacing in the corporate bond market. The
following is a stylized example of common situations occurring at that time.
EXAMPLE 1.3
Consider the following situation. There are two firms, Firm A and Firm B. Firm A
wants to raise M =$10 million using fixed rate coupon bonds, while Firm B wants
to raise M =$10 million using floating rate coupon bonds. Let the market rates
available to the two firms be the ones in Table 1.7. That is, Firm A can either borrow
at a fixed coupon rate of 15% or at a floating rate with coupon linked to the six-month
LIBOR rate plus 3%. Firm B, instead, can borrow fixed at 12% or floating at LIBOR
+ 2%. Note that the rates available to B are always lower than the ones available to
A, reflecting a difference in credit risk.
An investment bank observing the rates in Table 1.7 may offer the following deal
to the two firms.
First, Firm A issues a floating rate bond at LIBOR + 3%, while Firm B issues a
fixed rate bond at 12%. Second, the two firms swap coupon payments. In particular,
they could consider the following swap deal:
• Firm A pays B a fixed rate payment at 11% per year; and
• Firm B pays A a floating rate payment at LIBOR.
24
AN INTRODUCTION TO FIXED INCOME MARKETS
Figure 1.8
Firm A
A Swap Deal
Swap: 11%
=⇒=⇒=⇒
⇐=⇐=⇐=
Swap: LIBOR
Firm B
⇓
⇓
⇓
⇓
⇓
⇓
Bond
Market
Bond
Market
Bond A:
coupon =
LIBOR + 3%
Bond B:
coupon = 12%
Consider now the net cash flow from each of these firms when we take together
the bond issuance and the swap deal. For every future coupon date, we have:
Firm A Pays
Firm B Pays
(LIBOR + 3%) + 11% − LIBOR = 14%
to market
swap deal
:
12%
−11% + LIBOR = LIBOR + 1%
:
Figure 1.8 shows the cash flows in every period. Overall, we observe that Firm A
pays 14% instead of 15%, which was its fixed coupon market rate. Similarly, Firm
B pays LIBOR +1% instead of LIBOR +2%, which was its market rate. Both firms
then gain from entering this deal.
The early swaps managed to arbitrage some relative price discrepancy that existed
between floating rates and fixed rates. The notion was that if a spread exists because
of the risk of default, it should be the same across floating and fixed coupon bonds.
The difference in spreads between the two firms across asset classes, floating and
fixed coupon bonds, generates the possibility of a trade. How can we compute the
gains from a swap trade? The size of the pie that the two firms can divide through
a swap is given by the difference in comparative advantage implicit in Table 1.7. In
other words, the spread on the fixed coupon is 3%(= 15% − 12%) while the spread
on floating coupon is only 1%(= (LIBOR +3%) − (LIBOR +1%)). The difference
in spreads provide the total gains that can be split between the two firms:
Gains from trade = Fixed spread − Floating spread = 3% − 1% = 2%
(1.2)
In the example we divided the gains from trade equally between the two firms. In
reality, the exact split depends on the relative contractual strength of each firm: Firms
with higher creditworthiness would tend to get a higher coupon.
In addition, some part of the gain would also accrue to the investment bank that
brokers the deal.
The initial spur to swap trading was due to exploit arbitrage opportunities. At that time,
investment banks would also reap substantial profits from relatively large spreads. However,
as we will discuss in more detail in Chapter 5 and elsewhere, the growth in the swap
THE DERIVATIVES MARKET
25
market came about because of the extreme usefulness of swaps as a convenient means
for cash management and risk management. Financial institutions, corporations, and even
governments use swaps (a) to change the sensitivity of their cash flows to fluctuations in
interest rates; (b) to alter the timing of their payments and revenues; (c) or even simply for
investment purposes within complex trading strategies.
1.6.2
Futures and Forwards
Table 1.2 also reports the quotes of futures contracts, for instance, the “90 Day Eurodollar
Futures,”Eurodollar,futures the “Fed Funds Futures,” and the “10-year Treasury Note
Futures.” Futures contracts, discussed in detail in Chapter 6, are contracts according
to which two counterparties decide to exchange a security, or cash, or a commodity, at a
prespecified time in the future for a price agreed upon today. The quote represents the
price at which delivery will take place in the future. For interest rate futures, such as
Eurodollar or Fed funds, the quoted “price” is given by “100− futures rate.” For instance,
the 90-day Eurodollar futures contract with maturity “December,” quoted at 95.07 in Table
1.2, establishes the rate today, 4.93% = 100 − 95.07, at which the party long the futures
could deposit dollars in the Eurodollar market in December for the following 90 days.6
The futures market thus provides a convenient way for market participants to lock-in a
future interest rate: For instance, a corporation that has a large receivable due in December
can exploit the futures market to lock in the rate (4.93%) at which it can park the sum of
money for the following 90 days. In addition, the futures market is often used by market
participants to gauge the market expectation about future movement in interest rates.
Important in futures contracts is the fact that either counterparty may be called to make
payments in the future.
Table 1.2 does not report quotes of forward contracts, although we see from Table
1.1 that forward contracts make up a sizable share of the fixed income market. Forward
contracts are similar to futures contracts, in that two counterparties agree today that they
will exchange a security (or cash) in the future at a price that is decided today. Just like
futures contracts, forward contracts allow institutions to lock in interest rates for the future.
Unlike futures, forward contracts are not traded on regulated exchanges but only on the
over-the-counter market. Chapters 5 and 6 describe these contracts and delve into the
differences between futures and forwards.
1.6.3
Options
Table 1.2 does not provide any quotes for interest rate options, as we obtained this table
from the BTTM screen from a Bloomberg terminal, which only reports U.S. Treasury and
money market rates. But options contracts are a vital part of the fixed income market.
Table 1.1 shows that indeed the options’ market is quite larger than the futures and forward
markets. In addition, options are implicitly embedded in several other securities, such as
callable bonds, mortgage backed securities, and other types of structured notes. But first
of all:
What is an option?
6 In
fact, this futures is cash settled, so the deposit does not actually need to take place. See Chapter 6.
26
AN INTRODUCTION TO FIXED INCOME MARKETS
Intuitively, an option is the financial equivalent of an insurance contract: It is a contract
according to which the option buyer, who purchases the insurance, receives a payment from
the option’s seller, who sold the insurance, only if some interest rate scenario occurs in the
future. For instance, a corporation that issues a floating rate bond – a bond whose coupon
is tied to the level of a short-term floating rate – may be worried about an interest rate hike
in the future, a scenario that may drain too much financial resources from the coporation.
The corporation may purchase insurance against such scenarios, by purchasing a financial
option, called a cap, that pays only if the floating reference interest rate increases above
some cutoff point, called the strike rate. This contract would be a good hedge for the firm
against interest rate hikes, because, if the interest rate does increase above the strike rate,
then the option’s seller must pay the corporation a contractually agreed-upon cash flow,
which the corporation can use to pay its own liability to its bond holders.
Many options are implicit in many securities. A homeowner who financed the purchase
of his or her home using a adjustable rate mortgage (ARM), for instance, most likely also
bought (probably unknowingly) an option against an increase in interest rates. The reason
is that standard adjustable rate mortgages contain a provision stating that the maximum
rate the homeowner will have to pay over the life of the mortgage is capped at some level.
Therefore, the loan contract is equivalent to a standard floating rate loan contract plus an
option that pays if interest rates become too high, just like in the example of the corporation
above. Similarly, a homeowner who financed the purchase of his or her home using a
fixed rate mortgage also bought an option to pay back the mortgage whenever he or she
likes. In particular, homeowners pay back loans when the interest rate declines. The bank
making the mortgage implicitly sold the option to the homeowner, and the option premium
is embedded in the mortgage rate. Considering that the mortgage backed securities market
has become the dominant fixed income market in the U.S. (its value as of December 2008
is about $9 trillion, compared to only $6 trillion of the U.S. debt), the understanding of the
impact of options on fixed income instruments has never been more important.
1.7 ROADMAP OF FUTURE CHAPTERS
In this chapter we described some of the major fixed income markets. Starting with the next
chapter, we begin to analyze each market in much more detail. In Chapter 2 we cover the
basics of fixed income instruments, that is, the notion of a discount, of an interest rate, and
how we compute the fair valuation of Treasury bills, notes, and bonds. At the end of the
chapter we also show how we can use this information to obtain the price of some simple
structured securities, such as inverse floaters, which are popular securities if an investor
wishes to bet on a decrease in interest rates. Fixed income securities present many risks
for investors, even if they are issued or guaranteed by the U.S. government and therefore
they are default free. Indeed, long-term bonds, for instance, may suffer strong capital
losses in response to a generalized increase in interest rates. Chapters 3 and 4 discuss the
types of risk embedded in fixed income securities, the issue of risk measurement, as well
as the practice of risk management, such as asset-liability management and immunization
strategies. Chapters 5 and 6 cover popular fixed income derivatives, such as forward rates,
swaps, futures, and options, and their uses by market participants. Chapter 7 links the
fixed income market to the real economy. In particular, we talk about monetary policy,
economic growth, and inflation. In this context, we also discuss the market for TIPS, the
ROADMAP OF FUTURE CHAPTERS
27
inflation-protected debt securities. Finally, Chapter 8 discusses the residential mortgage
backed securities market, in terms of the types of securities as well as their riskiness from
an investment perspective. This chapter concludes the first part of the book, which aims at
providing some basic notions of fixed income securities.
Chapter 9 begins the second part of the book, which concerns the fair valuation of
derivative securities by no arbitrage. In particular, we begin with simple, one-period
binomial trees to explain the relations that have to exist between any pair of fixed income
securities. In this chapter we also introduce a popular pricing methodology called risk
neutral pricing. Chapter 10 expands the concepts of one-period binomial trees to multiple
periods, and discusses the issue of dinamic hedging, the standard methodology of hedging
a risk exposure by rebalancing the portfolio over time as the interest rate changes. Chapter
11 applies the methodology illustrated in the two earlier chapters to real-world securities.
In particular, it covers some popular models for the pricing of fixed income instruments, as
well as their estimation using real data. These concepts are further developed in Chapter
12, which details the pricing and hedging of an important class of derivatives, called
American options. Such options are implicitly embedded in numerous debt securities,
from callable bonds to mortgage backed securities. Chapter 13 illustrates a powerful
methodology for valuing and hedging complicated securities, namely, the Monte Carlo
simulations methodology. This methodology involves using computers to simulate interest
rate paths and price paths and then using those simulated quantities to compute current
prices and hedge ratios. We apply this methodology to real world securities, such as
corridor notes, ammortizing index swaps, mortgage backed securities, and collateralized
debt obbligations. This chapter concludes the second part of the book.
The third part of the book is mathematically more advanced, and some familiarity with
advanced calculus is required. In particular, Chapter 14 introduces continuous time methodologies, the notion of a Brownian motion, and Ito’s lemma. We apply these continuous time
methodologies and the rules of no arbitrage in Chapter 15 to compute the fair valuation of
Treasury notes and bonds, as well as derivative securities, such as options. Compared to
the second part of the book, which also accomplishes similar goals, the concepts discussed
in this part of the text are more realistic, and moreover provide analytical formulas for the
pricing and hedging of numerous securities, a very convenient property for traders who are
pressed for time. Chapter 16 discusses the notion of dynamic hedging, that is, the practice
by market participants of frequently rebalancing their portfolios to hedge their risk exposure. Chapter 17 introduces the notion of risk neutral pricing in continuous time models.
In addition, we develop for this class of models the Monte Carlo simulation approach to
pricing and hedging securities, a methodology widely used by market participants. In wellfunctioning markets, any risk embedded in fixed income securities should be compensated
for by a risk premium on its rate of return, either through a high coupon or a low purchasing
price. The link between risk and return of fixed income securities is the object of Chapter
18. In particular, we discuss the fact that if a security is providing an above-market coupon
or rate of return, then most likely this security is exposed to some risk, which perhaps is not
made completely explicit. To make this point clearer, we discuss a famous case involving
a special swap between Procter & Gamble and the investment bank Bankers Trust. Finally,
Chapters 19 to 22 cover more advanced models for fixed income security pricing. These
chapters discuss several examples in which models are applied to real world securities, and
draw some distinctions among them.
28
AN INTRODUCTION TO FIXED INCOME MARKETS
1.8 SUMMARY
In this chapter we covered the following topics:
1. Arbitrage strategies: These are strategies that cost nothing to enter into, and provide
sure money within a certain time. In well-functioning markets we should not expect
arbitrage strategies to persist for a long time. Indeed, pure arbitrage strategies are
rare in the market. The rules of no arbitrage determine the relative pricing across
fixed income securities and explain their high correlation.
2. U.S. Treasury market: The U.S. issues four types of securities: short-maturity T-bills,
medium-maturity T-notes, long-maturity T-bonds, and TIPS, the inflation-protected
securities. The size of the U.S. debt market is no longer dominant in fixed income
markets, as other markets became even larger, notably the mortgage backed securities
market and the derivatives market.
3. Money markets and money markets rates: This market is the source of short-term
borrowing by financial and non financial institutions. The main money market rates
are the commercial paper rate, LIBOR, and the Federal funds rate. The LIBOR rate,
the rate at which banks in London borrow from each other on an uncollateralized
basis, is the main reference rate in numerous derivative securities.
4. Repurchase agreements and the repo rate: In collateralized borrowing between two
counterparties, the repo rate is the borrowing or lending rate within a repurcahse
agreement. Because the borrowing is collateralized, the rate is lower than the LIBOR
rate, for instance.
5. Mortgage backed securities market: This is the largest debt market in United States.
Mortgage backed securities are collateralized by pools of residential and non residential mortgages and sold to investors who then receive claims to the mortgages
coupons. These securities present numerous additional risks for investors compared
to Treasury securities.
6. Swaps market: A swap is a contract according to which two counterparties agree to
exchange cash flows in the future. This market is very large in size, and although
considered a derivative market, its sheer size makes it equivalent to a primary market,
in the sense that the prices of swaps are really not derived from those of other
securities, but rather they depend on the relative size of demand and supply of these
contracts by market participants.
CHAPTER 2
BASICS OF FIXED INCOME SECURITIES
2.1
DISCOUNT FACTORS
Receiving a dollar today is not the same as receiving it in a month or in a year. There are
numerous reasons why people would like to have money today rather than in the future. For
one, money today can be put in a safe place (a bank or under the mattress) until tomorrow,
while the opposite is not easily doable. That is, money in hands gives its holder the option
to use it however he or she desires, including transferring it to the future through a deposit
or an investment. This option has a value on its own. If we agree that households and
investors value $1 today more than $1 in the future the question is then how much $1 in the
future is worth in today’s money. The value of what $1 in the future would be in today’s
money is called the discount factor. The notion of discount factors is at the heart of fixed
income securities.
It is easiest to introduce the concept by looking at a concrete example. The U.S.
government, as with most governments, needs to borrow money from investors to finance
its expenses. As discussed in Chapter 1, the government issues a number of securities, such
as Treasury bills, notes, and bonds, to investors, receiving money today in exchange for
money in the future. The U.S. Treasury is extremely unlikely to default on its obligations,
and thus the relation between purchase price and payoff of U.S. Treasury securities reveals
the market time value of money, that is, the exchange rate between money today and money
in the future. Example 2.1 illustrates this point.
29
30
BASICS OF FIXED INCOME SECURITIES
EXAMPLE 2.1
On August 10, 2006 the Treasury issued 182-day Treasury bills. The issuance market
price was $97.477 for $100 of face value.1 That is, on August 10, 2006, investors were
willing to buy for $97.477 a government security that would pay $100 on February 8,
2007. This Treasury bill would not make any other payment between the two dates.
Thus, the ratio between purchase price and the payoff, 0.97477 = $97.477/$100,
can be considered the market-wide discount factor between the two dates August 10,
2006 and February 8, 2007. That is, market participants were willing to exchange
0.97477 dollars on the first date for 1 dollar six months later.
Definition 2.1 The discount factor between two dates, t and T , provides the term of
exchange between a given amount of money at t versus a (certain) amount of money at a
later date T . We denote the discount factor between these two dates by Z(t, T ).
In the above example, the two dates are t = August 10, 2006 and T = February 08,
2007. The discount factor is Z (t, T ) = 0.97477.
In short, the discount factor Z(t, T ) records the time value of money between t and T .
Since it is a value (what is the value today of $1 in the future), it is essentially a price,
describing how much money somebody is willing to pay today in order to have $1 in the
future. In this sense, the notion of a discount factor is un-ambiguous. In contrast, as we
shall see below, the related notion of an interest rate is not un-ambiguous, as it depends
on compounding frequency, for instance. Exactly because discount factors unambiguously
represent a price – an exchange rate between money today versus money tomorrow – they
are at the heart of fixed income securities analysis. In the following sections we describe
their characteristics in more detail.
2.1.1 Discount Factors across Maturities
Definition 2.1 and Example 2.1 highlight that the discount factor at some date t (e.g., August
10, 2006) depends on its maturity T (e.g., February 8, 2007). If we vary the maturity T ,
making it longer or shorter, the discount factor varies as well. In fact, for the same reason
that investors value $1 today more than $1 in six months, they also value $1 in three months
more than $1 in six months. This can be seen, once again, from the prices of U.S. Treasury
securities.
EXAMPLE 2.2
On August 10, 2006 the U.S. government also issued 91-day bills with a maturity date
of November 9, 2006. The price was $98.739 for $100 of face value. Thus, denoting
again t = August 10, 2006, now T1 = November 9, 2006, and T2 = February 8,
2007, we find that the discount factor Z(t, T1 ) = 0.98739, which is higher than
Z(t, T2 ) = 0.97477.
1 These
2006.
data are obtained from the Web site http://www.treasurydirect.gov/RI/OFBills, accessed on August 22,
DISCOUNT FACTORS
31
This example highlights an important property of discount factors. Because it is always the
case that market participants prefer $1 today to $1 in the future, the following is true:
Fact 2.1 At any given time t, the discount factor is lower, the longer the maturity T . That
is, given two dates T1 and T2 with T1 < T2 , it is always the case that
Z(t, T1 ) ≥ Z(t, T2 )
(2.1)
The opposite relation Z(t, T1 ) < Z(t, T2 ) would in fact imply a somewhat curious
behavior on the part of investors. For instance, in the example above in which T1 =
November 9, 2006 and T2 = February 8, 2007, if Z(t, T1 ) was lower than Z(t, T2 ) =
0.97477, it would imply that investors would be willing to give up $97.477 today in order
to receive $100 in six months, but not in order to receive the same amount three months
earlier. In other words, it implies that investors prefer to have $100 in six months rather
than in three months, violating the principle that agents prefer to have a sum of money
earlier rather than later. Moreover, a violation of Relation 2.1 also generates an arbitrage
opportunity, which we would not expect to last for long in well functioning financial markets
(see Exercise 1). In Chapter 5 we elaborate on this topic, showing also that a violation of
Relation 2.1 amounts to the assumption that future nominal interest rates be negative.
2.1.2
Discount Factors over Time
A second important characteristic of discount factors is that they are not constant over time,
even while keeping constant the time-to-maturity T − t, that is, the interval of time between
the two dates t and T in the discount factor Z(t, T ). As time goes by, the time value of
money changes. For instance, the U.S. Treasury issued a 182-day bill on t1 = August 26,
2004, with maturity T2 = February 24, 2005, for a price of $99.115. This price implies a
discount factor on that date equal to Z(t1 , T1 ) = 0.99115. This value is much higher than
the discount factor with the same time to maturity (six months) two years later, on August
10, 2006, which we found equal to 0.97477.
Figure 2.1 plots three discounts factors over time, from January 1953 to June 2008.2 The
top solid line is the 3-month discount factor, the middle dotted line is the 1-year discount
factor, and bottom dashed line is the 3-year discount factor. First, note that indeed on each
date in the sample, the discount factor with shorter time to maturity is always higher than
the discount factor with longer time to maturity. Second, the variation of discount factors
over time is rather substantial. For instance, the 3-year discount factor is as low as 0.6267
in August 1981, and as high as 0.95 in June 1954 and in June 2003.
Why do discount factors vary over time? Although this is a topic of a later chapter,
it is useful to provide here the most obvious, and intuitive, reason. Figure 2.2 plots the
time series of expected inflation from 1953 to 2008.3 Comparing the discount factor series
2 Data
excerpted from CRSP ( Fama Bliss discount bonds) ¤2009 Center for Research in Security Prices (CRSP),
The University of Chicago Booth School of Business. We discuss methodologies to estimate discount factors
from bond data in Section 2.4.2 and in the Appendix.
3 The expected inflation series is computed as the predicted annual inflation rate resulting from a rolling regression
of inflation on its 12 lags. We present more details in Chapter 7.
32
BASICS OF FIXED INCOME SECURITIES
Figure 2.1
Discount Factors
100
95
Discount Factor (%)
90
85
80
75
70
3 months
65
1 year
3 years
60
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
Source: Center for Research in Security Prices (CRSP)
plotted in Figure 2.1 with the expected annual inflation series in Figure 2.2 it appears that
expected inflation is an important determinant of discount factors. The intuition is also
quite straightforward: Inflation is exactly what determines the time value of money, as it
determines how much goods money can buy. The higher the expected inflation, the less
appealing it is to receive money in the future compared to today, as this money will be able
to buy a lesser amount of goods.
Although expected inflation is the most obvious culprit in explaining the variation over
time of discount factors, it is not the only one. In Chapter 7 we look at various explanations
that economists put forward to account for the behavior of discount factors and interest
rates. These explanations are related to the behavior of the U.S. economy, its budget
deficit, and the actions of the Federal Reserve, as well as investors’ appetite for risk (or
lack thereof). These macro economic conditions affect the relative supply and demand of
Treasury securities and thus their prices.
2.2 INTEREST RATES
Grasping the concept of a rate of interest is both easier and more complicated than absorbing
the concept of a discount factor. It is easier because the idea of interest is closer to our
everyday notion of a return on an investment, or the cost of a loan. For instance, if we invest
$100 for one year at the rate of interest of 5%, we receive in one year $105, that is, the
original capital invested plus the interest on the investment. The same investment strategy
could be described in terms of a discount factor as well: The discount factor here is the
INTEREST RATES
33
Figure 2.2 Expected Inflation
16
14
Expected Inflation Rate (%)
12
10
8
6
4
2
0
−2
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
Data Source: Bureau of Labor Statistics.
exchange rate between having $105 in one year or $100 today, that is 0.9524 = $100/$105.
This latter number, which is equivalent to the 5% rate of interest, perhaps less intuitively
describes the return on an investment.
The concept of an interest rate, however, is also more complicated, because it depends on
the compounding frequency of the interest paid on the initial investment. The compounding
frequency is defined as follows:
Definition 2.2 The compounding frequency of interest accruals refers to the annual
number of times in which interest is paid and reinvested on the invested capital.
Indeed, to some extent, mentioning only an interest rate level is an incomplete description
of the rate of return of an investment, or the cost of a loan or mortgage. The compounding
frequency is a crucial element that must be attached to the interest rate figure.
For instance, in the above example, we implicitly assumed that the 5% rate of interest is
applied to the original capital only once (and hence the $105 result). However, if interest
accrues, say, every 6 months, then the correct amount at maturity would be
5%
5%
× 1+
= $105.0625,
($100) × 1 +
2
2
which is higher than $105. If interest accrues every month, then the correct amount at
maturity would be
5%
5%
× . . . [ 12 times ] . . . × 1 +
= $105.11,
($100) × 1 +
12
12
34
BASICS OF FIXED INCOME SECURITIES
which is higher still. This example shows the following:
Fact 2.2 For a given interest rate figure (e.g. 5%), more frequent accrual of interest yields
a higher final payoff.
Looking at interest rates from a different perspective, consider an investment in a
security that costs $100 today and that pays $105 in one year, as in our earlier example.
What is the rate of interest on this security? The intuitive answer is 5%, because we
invest $100 and obtain $105, and thus the return equals 5% = ($105 − $100)/$100.
However, once again, the correct answer really depends on the frequency with which
interest accrues on the security. If the interest accrues once a year, then 5% is the correct
answer. However, if the interest accrues every half a year, for instance, the correct answer
is given by r = 4.939%. In fact, ($100) × (1 + 4.939%/2) × (1 + 4.939%/2) = $105,
which is indeed the payoff from the investment of $100. Similarly, if the interest accrues
every month, then the correct answer to the same question is given by r = 4.89%, as
($100) × (1 + 4.89%/12) × . . . [ 12 times ] . . . × (1 + 4.89%/12) = $105.
Fact 2.3 For a given final payoff, more frequent accrual of interest implies a lower interest
rate figure.
This discussion emphasizes also the crucial difference between a rate of return on an
investment, and an interest rate, which are related, but different, concepts. The rate of
return is indeed the difference between payoff and initial investment, divided by the latter.
In the example, 5% = ($105 − $100)/$100 is the rate of return on the investment. The
rate of interest corresponds instead to the (annualized) rate of return on the investment
within the compounding period, but it differs from it otherwise. For instance, if the rate
of interest is 5% and it accrues semi-annually, then within a six-month period the rate of
return on the investment is 2.5%, that is, from $100 we have in six months $102.5. If we
annualize this semi-annual return we obtain 5%, which corresponds to the rate of interest.
Note, however, the rate of interest and the rate of return differ for a one-year horizon. In
one year, the original investment will pay $105.0625, as we obtained earlier, and thus the
rate of return is 5.0625%>5%. When the horizon is longer, the discrepancy between the
annualized interest rate figure and the annualized rate of return on the investment is larger.
2.2.1 Discount Factors, Interest Rates, and Compounding Frequencies
The examples above illustrate that discount factors and interest rates are intimately related,
once we make explicit the compounding frequency. Given an interest rate and its compounding frequency, we can define a discount factor. Vice versa, given a discount factor,
we can define an interest rate together with its compounding frequency. In this section we
make the relation explicit.
Two compounding frequencies are particularly important: semi-annual compounding
and continuous compounding. The semi-annual compounding frequency is the standard
benchmark, as it matches the frequency of coupon payments of U.S. Treasury notes and
bonds. The continuous compounding, defined below, is also important, mainly for its
analytical convenience. As we shall see, formulas and derivations are much simpler under
the assumption that the interest on an investment accrues infinitely frequently. This is of
course an abstraction, but a useful one.
35
INTEREST RATES
2.2.1.1 Semi-annual Compounding Let’s begin with an example:
EXAMPLE 2.3
Let t = August 10, 2006, and let T = August 10, 2007 (one year later). Consider
an investment of $100 at t at the semi-annually compounded interest r = 5%, for
one year. As mentioned earlier, this terminology means that after six months the
investment grows to $102.5 = $100 × (1 + 5%/2), which is then reinvested at the
same rate for another six months, yielding at T the payoff:
Payoff at T = $105.0625 = ($100) × (1 + r/2) × (1 + r/2) = ($100) × (1 + r/2)2
Given that the initial investment is $100, there are no cash flows to the investor during
the period, and the payoff at T is risk free, the relation between money at t ($100)
and money at T (= $105.0625 = payoff at T ) establishes a discount factor between
the two dates, given by
Z(t, T ) =
1
$100
=
payoff at T
(1 + r/2)2
This example underlies the following more general statement:
Fact 2.4 Let r2 (t, T ) denote the (annualized) semi-annually compounded interest rate
between t and T . Then r2 (t, T ) defines a discount factor as
Z(t, T ) = 1
1+
r 2 (t,T )
2
2×(T −t)
(2.2)
The logic of this fact lies in the example above. The semi-annually compounded interest
rate r2 (t, T ) defines a payoff at maturity T given by
2×(T −t)
r2 (t, T )
Payoff at T = Investment at t × 1 +
.
2
Since the payoff at T is known at t, the relation between investment today at t and the
payoff at T defines the time value of money, and Z(t, T ) given in Equation 2.2 defines the
rate of exchange between money at T and money at t.
Similarly, given a discount factor Z(t, T ), we can obtain the semiannually compounded
interest rate. The following example illustrates the point.
EXAMPLE 2.4
On March 1, 2001 (time t) the Treasury issued a 52-week Treasury bill, with maturity
date T = February 28, 2002. The price of the Treasury bill was $95.713. As we
have learned, this price defines a discount factor between the two dates of Z(t, T ) =
0.95713. At the same time, it also defines a semi-annually compounded interest
rate equal to r2 (t, T ) = 4.43%. In fact, $95.713 × (1 + 4.43%/2)2 = $100. The
36
BASICS OF FIXED INCOME SECURITIES
semi-annually compounded interest rate can be computed from Z(t, T ) = 0.95713
by solving for r2 (t, T ) in Equation 2.2:
r2 (t, T ) = 2 ×
1
1
Z (t, T ) 2
−1
=2×
1
1
0.95713 2
− 1 = 4.43%
(2.3)
Fact 2.5 Let Z(t, T ) be the discount factor between dates t and T . Then the semi-annually
compounded interest rate r2 (t, T ) can be computed from the formula
r2 (t, T ) = 2 ×
1
1
Z (t, T ) 2 ×( T −t )
−1
(2.4)
2.2.1.2 More Frequent Compounding Market participants’ time value of money
– the discount factor Z(0, T ) – can be exploited to determine the interest rates with any
compounding frequency, as well as the relation that must exist between any two interest
rates which differ in compounding frequency. More precisely, if we let n denote the number
of compounding periods per year (e.g., n = 2 corresponds to semi-annual compounding),
we obtain the following:
Fact 2.6 Let the discount factor Z(t, T ) be given, and let rn (t, T ) denote the (annualized)
n-times compounded interest rate. Then rn (t, T ) is defined by the equation
Z(t, T ) = 1
1+
r n (t,T )
n
n ×(T −t)
(2.5)
Solving for rn (t, T ), we obtain
rn (t, T ) = n ×
1
1
Z (t, T ) n ×( T −t )
−1
(2.6)
For instance, a $100 investment at the monthly compounded interest rate r12 (0, 1) = 5%
yields by definition
r12 (0, 1)
r12 (0, 1)
Payoff at T = $100× 1 +
×· · · [ 12 times ] · · ·× 1 +
= $105.1162
12
12
Thus, the monthly compounded interest rate r12 (0, 1) = 5% corresponds to the discount
factor Z(0, 1) = $100/$105.1162 = 0.95133, and vice versa.
2.2.1.3 Continuous Compounding. The continuously compounded interest rate
is obtained by increasing the compounding frequency n to infinity. For all practical
purposes, however, daily compounding – the standard for bank accounts – closely matches
the continuous compounding, as we see in the next example.
37
INTEREST RATES
Table 2.1
Interest Rate and Compounding Frequency
n
rn (t, t + 1)
Annual
Semi-annual
Monthly
Bi-monthly
Weekely
Bi-weekly
Daily
Bi-daily
Hourly
1
2
12
24
52
104
365
730
8760
5.000%
4.939%
4.889%
4.883%
4.881%
4.880%
4.879%
4.879%
4.879%
Continuous
∞
4.879%
Compounding Frequency
EXAMPLE 2.5
Consider the earlier example in which at t we invest $100 to receive $105 one year
later. Recall that the annually compounded interest rate is r1 (t, t + 1) = 5%, the
semi-annually compounded interest rate is r2 (t, t + 1) = 4.939%, and the monthly
compounded interest rate is r12 (t, t + 1) = 4.889%. Table 2.1 reports the n−times
compounded interest rate also for more frequent compounding. As it can be seen,
if we keep increasing n, the n− times compounded interest rate rn (t, t + 1) keeps
decreasing, but at an increasingly lower rate. Eventually, it converges to a number,
namely, 4.879%. This is the continously compounded interest rate. Note that in this
example, there is no difference between the daily compounded interest rate (n = 252)
and the one obtained with higher frequency (n > 252). That is, we can mentally
think of continuous compounding as the daily compounding frequency.
Mathematically, we can express the limit of rn (t, T ) in Equation 2.6 as n increases to
infinity in terms of the exponential function:
Fact 2.7 The continuously compounded interest rate r(t, T ), obtained from rn (t, T ) for
n that increases to infinity, is given by the formula
Z(t, T ) = e−r (t,T )(T −t)
(2.7)
ln (Z(t, T ))
T −t
(2.8)
Solving for r(t, T ), we obtain
r (t, T ) = −
where “ln(.)” denotes the natural logarithm.
Returning to Example 2.5, we can verify Equation 2.8 by taking the natural logarithm
of Z(t, T ) = $100/$105 = .952381 and thus obtaining
r(t, T ) = −
ln(Z(t, t + 1))
= 4.879%
1
38
BASICS OF FIXED INCOME SECURITIES
2.2.2 The Relation between Discounts Factors and Interest Rates
The previous formulas show that given a discount factor between t and T , Z(t, T ), we
can define interest rates of any compounding frequency by using Equation 2.2, 2.5, or 2.7.
This fact implies that we can move from one compounding frequency to another by using
the equalities implicit in these equations. For instance, for given interest rate rn (t, T ) with
n compounding frequency, we can determine the continuously compounded interest rate
r(t, T ) by solving the equation
e−r (t,T )(T −t) = Z(t, T ) = 1
1+
r n (t,T )
n
n ×(T −t)
(2.9)
Because of its analytical convenience, in this text we mostly use the continuously compounded interest rate in the description of discount factors, and for other quantities. Translating such a number into another compounding frequency is immediate from Equation 2.9,
which, more explicitly, implies
rn (t, T )
r(t, T ) = n × ln 1 +
(2.10)
n
r (t ,T )
(2.11)
rn (t, T ) = n × e n − 1
To conclude, then, this section shows that the time value of money can be expressed
equivalently through a discount factor, or in terms of an interest rate with its appropriate
compounding frequency. At times, it will be convenient to focus on discount factors and
at other times on interest rates, depending on the exercise. We should always keep in mind
that the two quantities are equivalent.
2.3 THE TERM STRUCTURE OF INTEREST RATES
In the previous sections we noted that the primitive of our analysis is the discount factor,
from which we define interest rates of various compounding frequencies. Interest rates,
though, have a big advantage over discount factors when we analyze the time value of
money: their units can be made uniform across maturities by annualizing them. The
following example illustrates the point.
EXAMPLE 2.6
On June 5, 2008, the Treasury issued 13-week, 26-week and 52-week bills at prices
$99.5399, $99.0142, and $97.8716, respectively. Denoting t = June 5, 2008, and
T1 , T2 , and T3 the three maturity dates, the implied discount factors are Z(t, T1 ) =
0.995399, Z(t, T2 ) = 0.990142, and Z(t, T3 ) = 0.978716. The discount factor
of longer maturities is lower than the one of shorter maturities, as Fact 2.1 would
imply. The question is then: How much lower is Z(t, T3 ), say, compared to Z(t, T2 )
or Z(t, T1 )? Translating the discount factors into annualized interest rates provides
a better sense of the relative value of money across maturities. In this case, the
continuously compounded interest rates are
r(t, T1 ) = −
ln(0.995399)
= 1.8444%;
0.25
THE TERM STRUCTURE OF INTEREST RATES
r(t, T2 )
r(t, T3 )
39
ln(0.990142)
= 1.9814%;
0.5
ln(0.978716)
= 2.1514%.
= −
1
= −
The time value of money rises with maturity: The compensation that the Treasury
has to provide investors to make them part with money today to receive money in the
future, i.e., hold Treasury securities, increases the longer the investment period.
The term structure of interest rates is defined as follows:
Definition 2.3 The term structure of interest rates, or spot rate curve, or yield curve,
at a certain time t defines the relation between the level of interest rates and their time to
maturity T − t. The discount curve at a certain time t defines instead the relation between
the discount factors Z(t, T ) and their time to maturity T − t.
Figure 2.3 provides examples of spot curves r(t, T ) at four different dates.4 These dates
have been chosen also because the spot curves had different “shapes.” Traders refer to
these different shapes with particular names, which we now describe.5
Panel A of Figure 2.3 plots the term structure of interest rates on October 30, 1992. On
the horizontal axis we have time to maturity m = T − t for m that ranges from 3 months
(m = 0.25) to 10 years (m = 10) (the letter “m” stands for “maturity”). The vertical axis
represents the interest rate level r(t, t + m) that corresponds to the various maturities. As
can be seen, the term structure of interest rates on October 30, 1992 was increasing, which
is a typical pattern in the United States. The difference between the 10-year interest rate
and the short-term interest rate is about 4%. This difference is called the term spread, or
slope, of the term structure of interest rates.
Definition 2.4 The term spread, or slope, is the difference between long-term interest
rates (e.g. 10-year rate) and the short-term interest rates (e.g. 3-month rate).
Typically, in the U.S. the term spread is positive. How is the term spread determined?
Like discount factors, the term spread depends on numerous variables, such as expected
future inflation, expected future growth of the economy, agents’ attitudes toward risk, and
so on. It is worth mentioning that although the expectation of future higher interest rates
may determine today’s term structure of interest rates, this is not the only channel. We
will discuss more precisely the determinants of the term structure of interest rates in later
chapters.
The shape of the term structure of interest rates is not always increasing. Panels B - D
of Figure 2.3 plot the shape of the term structure on other occasions. In particular, Panel B
illustrates a decreasing term structure of interest rates, as occurred on November 30, 2000.
Panel C plots a term structure that is first rising and then decreasing. This shape is called
hump and, in the example, took place on March 31, 2000. Finally, Panel D plots a term
4 We
calculated the spot curves using the extended Nelson Siegel model (see Section 2.9.3 in the appendix at the
end of the chapter) based on data from CRSP (Monthly Treasuries) ¤2009 Center for Research in Security Prices
(CRSP), The University of Chicago Booth School of Business.
5 We are using the continuously compounded interest rate r(t, T ) to describe the curve. This is arbitrary. We
could use any other compounding frequency, but as mentioned earlier, the continuously compounded frequency
has some analytical advantages, as we shall see.
40
BASICS OF FIXED INCOME SECURITIES
Figure 2.3
The Shapes of the Term Structure
B: 11/30/2000 : Decreasing
6.4
7
6.2
Interest Rate (%)
Interest Rate (%)
A: 10/30/1992 : Increasing
8
6
5
4
5.8
5.6
3
2
6
0
2
4
6
Time to Maturity
8
5.4
10
0
4
6
Time to Maturity
8
10
D: 07/31/1989 : Inverted Hump
6.4
8
6.3
7.9
Interest Rate (%)
Interest Rate (%)
C: 03/31/2000 : Hump
2
6.2
6.1
6
5.9
7.8
7.7
7.6
7.5
0
2
4
6
Time to Maturity
8
10
7.4
0
2
4
6
Time to Maturity
8
10
Data Source: Center for Research in Security Prices
structure that is first decreasing and then increasing. This shape is called an inverted hump.
The example in the Panel D is for July 31, 1989.
2.3.1 The Term Structure of Interest Rates over Time
As for discount factors, the term structure of interest rates depends on the date t at which
it is computed. This is evident in Figure 2.3, as on three different dates we have three
different shapes of the term structure. In particular, the dates corresponding to Panel C and
Panel B of Figure 2.3 are only eight months apart, and yet the term structures are quite
different in shape. In addition, besides the change in shape, the term structure also moves
up and down as a whole.
THE TERM STRUCTURE OF INTEREST RATES
41
Different dates correspond to different term structure of interest rates. For instance,
Figure 2.4 plots the term structure of interest rates on three different dates, at six-months
intervals, namely, from January 31, 1994 to January 31, 1995. In all three cases, the term
structure of interest rates is increasing, but it is clear that it is lower and steeper for the first
date, while it is higher and flatter for the third date. The increase in the term structure is
rather substantial: The short-term rate passed from 2.9% to 5.5%, while the long rate passed
from 5.7% to 7.5%. This large change in the term structure of interest rates may have a
devastating effect on the value of portfolios heavily invested in fixed income instruments.
As we discuss in this and Chapter 3, it was is exactly in 1994 that Orange County, a rich
county in California, lost $1.6 billion and went bankrupt. The unexpected hike in interest
rates together with an aggressive leveraged investment portfolio were the main causes of
the debacle.
Figure 2.4 The Term Structure of Interest Rates on Three dates
8
7.5
7
6.5
Interest Rate (%)
6
5.5
5
4.5
4
3.5
01/31/1994
07/29/1994
3
01/31/1995
2.5
0
1
2
3
4
5
6
Time to Maturity
7
8
9
10
Data Source: Center for Research in Security Prices
As we did for discount factors, we can plot various points of the term structure of interest
rates over time. Figure 2.5 graphs the term structure with maturities 3 months, 1 year, and
5 years from 1965 to 2008.6 The first pattern that we see in Figure 2.5 is that interest rates
move up and down substantially. The second clear pattern is that all of the interest rates
move together: They go up and down at roughly the same time. However, it is also clear
that they do not move up and down by the same amounts. To see this, let us consider the
dotted line at the bottom of the plot, and the dashed line at the top. They correspond to the
3-month and 5-year interest rates, respectively. As can be seen, they both go up and down
6 Data
excerpted from CRSP (Fama Bliss Discount Bonds and Fama Risk Free Rate) ¤2009 Center for Research
in Security Prices (CRSP), The University of Chicago Booth School of Business.
42
BASICS OF FIXED INCOME SECURITIES
at roughly the same time. We can also see that the term spread, the distance between the
two rates, changes over time as well. For instance, focusing on the last part of the period,
2000 - 2006, we see that while the 3-month rate moved from over 6% down to 1%, then
went back up to 5%, the 5-year rate moved from 6% down to 3%, and then back up to 5%.
That is, the spread was zero both at the beginning and at the end of this sample, but it was
large in the middle.
Figure 2.5
The Term Structure over Time
0.18
3 months
1 year
5 years
0.16
0.14
Interest Rate (%)
0.12
0.1
0.08
0.06
0.04
0.02
0
1955
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
Source: Center for Research in Security Prices
Why does the term spread change over time? Once again, there are numerous reasons
that contribute to the variation of both interest rates and term spreads, such as fluctuations
in expected inflation, expected economic growth, and risk attitude of investors. We will
review some recent theories in Chapter 7
2.4 COUPON BONDS
U.S. government Treasury bills involve only one payment from the Treasury to the investor
at maturity. That is, their coupon rate is zero. They are thus an example of zero coupon
bonds, bonds with no intermediate cash flows between issue date and maturity. The
knowledge of the prices of zero coupon bonds allows us to determine the discount factor
Z(t, T ), as described in the previous sections. More specifically, a government zero coupon
bond at time t with maturity T has a price equal to
Pz (t, T ) = 100 × Z(t, T )
The subscript “z” is a mnemonic term for “zero” in zero coupon bond.
(2.12)
43
COUPON BONDS
The Treasury issues zero coupon bonds with maturities up to only 52 weeks. For
longer maturities, the Treasury issues securities that carry a coupon, that is, they also pay
a sequence of cash flows (the coupons) between issue date and maturity, in addition to the
final principal. In particular, the U.S. government issues Treasury notes, which are fixed
income securities with maturity up to 10 years; Treasury bonds, which have maturities up
to 30 years; and TIPS (Treasury Inflation Protected Securities), which have coupons that
are not constant, but rather are linked to a recent inflation rate figure. We will talk about
TIPS more exhaustively in Chapter 7. For now, we only consider Treasury notes and bonds.
For convenience, we refer to both types as coupon bonds.
2.4.1
From Zero Coupon Bonds to Coupon Bonds
In this section we establish a relation between the prices of zero coupon bonds and coupon
bonds. This relation forms the basis of much of the analysis that follows in later chapters,
and so it is particularly important.
First, note that a coupon bond can be represented by the sequence of its cash payments.
For instance, the 4.375% Treasury note issued on January 3, 2006 and with maturity of
December 31, 2007, pays a cash flow of $2.1875 on June 30, 2006, December 31, 2006,
and June 30, 2007, while it pays $102.1875 on December 31, 2007. Given the sequence of
cash flows, which are certain in the sense that the U.S. Government is extremely unlikely
to default, we could compute the value of the bond itself if we knew the discount factors
Z(t, T ) to apply to each of the four dates. In fact, we can discount each future cash flow
using its own discount factor, and sum the results.
Fact 2.8 Consider a coupon bond at time t with coupon rate c, maturity T and payment
dates T1 , T2 ,...,Tn = T . Let there be discount factors Z(t, Ti ) for each date Ti . Then the
value of the coupon bond can be computed as
Pc (t, Tn )
=
c × 100
c × 100
× Z(t, T1 ) +
× Z(t, T2 ) + ...
2
2
c × 100
+ 100 × Z (t, Tn )
... +
2
n
=
c × 100
×
Z(t, Ti ) + 100 × Z(t, Tn )
2
i= 1
=
c
×
Pz (t, Ti ) + Pz (t, Tn )
2 i= 1
(2.13)
n
(2.14)
The subscript “c” is a mnemonic device for “coupon” in coupon bond. Formula 2.14
shows that the coupon bond can be considered as a portfolio of zero coupon bonds.
EXAMPLE 2.7
Consider the 2-year note issued on t = January 3, 2006 discussed earlier. On this date,
the 6-month, 1-year, 1.5-years, and 2-year discounts were Z(t, t + 0.5) = 0.97862,
Z(t, t + 1) = 0.95718, Z(t, t + 1.5) = 0.936826 and Z(t, t + 2) = 0.91707.
44
BASICS OF FIXED INCOME SECURITIES
Therefore, the price of the note on that date was
4
Pc (t, Tn ) = $2.1875 ×
Z(t, t + 0.5 × i) + $100 × 0.91707 = $99.997,
i= 1
which was indeed the issue price at t.
We can also represent the value of the coupon bond by using the semi-annual interest rate
r2 (t, Ti ), where Ti , i = 1, ..., n, are the coupon payment dates. This representation is
derived from the basic one above, but it can be useful nonetheless to report it:
n
Pc (t, Tn )
=
i= 1
c/2 × 100
(1 + r2 (t, Ti ) /2)
2×(T i −t)
+
100
(1 + r2 (t, Tn ) /2)
2×(T n −t)
(2.15)
A useful fact is the following:
Fact 2.9 Let the semi-annual discount rate be constant across maturities, r2 (t, Ti ) = r2
for every Ti . At issue date t = 0, the price of a coupon bond with coupon rate equal to the
constant semi-annual rate c = r2 is equal to par.
n −1
Pc (0, Tn )
=
i= 1
c/2 × 100
(1 + r2 /2)
2×T i
+
100 × (1 + c/2)
(1 + r2 /2)
2×T n
= 100
(2.16)
To understand the above fact, consider a 1-year note. Then, we can write
Pc (0, T2 )
=
=
=
=
=
c/2 × 100 100 × (1 + c/2)
+
2
1 + r2 /2
(1 + r2 /2)
100
1 + c/2
c/2 × 100
+
×
1 + r2 /2
1 + r2 /2
1 + r2 /2
100
c/2 × 100
+
×1
1 + r2 /2
1 + r2 /2
100 × (1 + c/2)
1 + r2 /2
100
(2.17)
(2.18)
(2.19)
(2.20)
(2.21)
This argument can be extended to many periods. The intuition is that any additional periods
increase the cash flow by c/2 while they also increase the discount rate by the same amount
r2 /2. The two forces move in opposite directions (more cash flows imply higher prices,
while the additional discount implies lower price).
2.4.1.1 A No Arbitrage Argument We can establish Equation 2.13 also by appealing to a no arbitrage argument. In well-functioning markets in which both the coupon bond
Pc (t, Tn ) and the zero coupon bonds Pz (t, Ti ) are traded in the market, if Equation 2.13
did not hold, an arbitrageur could make large risk-free profits. For instance, if
c
c
c
Pc (t, Tn ) < × Pz (t, T1 ) + × Pz (t, T2 ) + ... + 1 +
(2.22)
× Pz (t, Tn )
2
2
2
COUPON BONDS
45
then the arbitrageur can buy the coupon bond for Pc (t, Tn ) and sell immediately c/2 units
of zero coupon bonds with maturities T1 , T2 ,..,Tn −1 and (1 + c/2) of the zero coupon bond
with maturity Tn . This strategy yields an inflow of money to the arbitrageur that is equal to
the difference between the right-hand side and the left-hand side of Equation 2.22. At every
other maturity Ti the arbitrageur has a zero net position, as he receives the coupon from the
Treasury and turns it around to the investors to whom the arbitrageur sold the individual
zero coupon bonds. We note that this reasoning is the one that stands behind the law of one
price, introduced in Fact 1.1 in Chapter 1, the fact that securities with identical cash flows
should have the same price. The following example further illustrates the concept.
EXAMPLE 2.8
In Example 2.7, suppose that the 2-year note was trading at $98. An arbitrageur
could purchase, say, $98 million of the 2-year note, and sell $2.1875 million of the
6-month, 1-year and 1.5-year zero coupons, and $102.1875 million of the 2-year zero
coupon bond. The total value of the zeros the arbitrageur sells is $99.997 million,
realizing approximately $2 million. The strategy is risk free, because at each coupon
date in the future, the arbitrageur receives $2.1875 million from the Treasury, which
he simply turns over to the investors who bought the zero coupon bonds. Similarly,
at maturity, the arbitrageur receives $102.1875 million from the Treasury, and again
turns it around to the investors of the last coupon.
In well-functioning markets such arbitrage opportunities cannot last for long. Thus, Equation 2.13 should hold “most of the time.” It may happen that due to lack of liquidity
or trading, some arbitrage opportunities may be detectable in the relative pricing of zero
coupon bonds, such as STRIPS, and coupon bonds. However, these arbitrage opportunities
are seldom exploitable: As soon as an arbitrageur tries to set up an arbitrage like the one
described above, prices move instantly and the profit vanishes. Because expert arbitrageurs
know this fact, some apparent mispricing may persist in the market place. We will regard
such situations as “noise,” that is, a little imprecision in market prices due to liquidity or
external factors that sometimes impede the smooth functioning of capital markets.
2.4.2
From Coupon Bonds to Zero Coupon Bonds
We can also go the other way around: If we have enough coupon bonds, we can compute
the implicit value of zero coupon bonds from the prices of coupon bonds. Equation 2.13
can be used to estimate the discount factors Z(t, T ) for every maturity. The following
example illustrates the reasoning:
EXAMPLE 2.9
On t = June 30, 2005, the 6-month Treasury bill, expiring on T1 = December 29,
2005, was trading at $98.3607. On the same date, the 1 year to maturity, 2.75%
Treasury note, was trading at $99.2343. The maturity of the latter Treasury note
is T2 = June 30, 2006. Given Equation 2.13, we can write the value of the two
46
BASICS OF FIXED INCOME SECURITIES
securities as:7
Pbill (t, T1 ) = $98.3607
=
$100 × Z(t, T1 )
Pn ote (t, T2 ) = $99.2343
=
$1.375 × Z(t, T1 ) + $101.375 × Z(t, T2 ) (2.24)
(2.23)
We have two equations in two unknowns [the discount factors Z(t, T1 ) and Z(t, T2 )].
As in Section 2.1, from the first equation we obtain the discount factor Z(t, T1 ) =
$98.3607/$100 = 0.983607. We can substitute this value into the second equation,
and solve for Z(t, T2 ) to obtain:
Z(t, T2 ) =
$99.2343 − $1.375 × 0.983607
$99.2343 − $1.375 × Z(t, T1 )
=
= 0.965542
$101.375
$101.375
The prices of coupon bonds, then, implicitly contain the information about the market time
value of money. This procedure can be iterated forward to obtain additional terms.
EXAMPLE 2.10
On the same date, t = June 30, 2005, the December 31, 2006 Treasury note, with
coupon of 3%, was trading at $99.1093. Denoting by T3 = December 31, 2006, the
price of this note can be written as
Pn ote (t, T3 ) = $99.1093
=
$1.5 × Z(t, T1 ) + $1.5 × Z(t, T2 ) + $101.5 × Z(t, T3 )
(2.25)
We already determined Z(t, T1 ) = 0.983607 and Z(t, T2 ) = 0.965542 in Example
2.9. In Equation 2.25 the only unknown element is Z(t, T3 ). This is one equation in
one unknown, and so we can solve for the Z(t, T3 ) to obtain
Z(t, T3 ) =
=
=
$99.1093 − $1.5 × (Z(t, T1 ) + Z(t, T2 ))
$101.5
$99.1093 − $1.5 × (0.983607 + 0.965542)
$101.5
0.947641
If we have a sufficient amount of data, we can proceed in this fashion for every maturity,
and obtain all of the discount factors Z(t, T ). This methodology is called the bootstrap
methodology.
Definition 2.5 Let t be a given date. Let there be n coupon bonds, with coupons ci ,
maturities Ti and prices denoted by P (t, Ti ). Assume that maturities are at regular
intervals of six months, that is, T1 = t + 0.5 and Ti = Ti−1 + 0.5. Then, the bootstrap
methodology to estimate discount factors Z(t, Ti ) for every i = 1, ..., n is as follows:
1. The first discount factor Z(t, T1 ) is given by
Z(t, T1 ) =
7 Notice
Pc (t, T1 )
100 × (1 + c1 /2)
(2.26)
a little approximation in this computation: The T-note would pay its coupon on December 31, 2005,
rather than December 29. We assume that both dates correspond, approximately, to T 1 .
COUPON BONDS
2. Any other discount factor Z(t, Ti ) for i = 2, ..., n is given by
i−1
Pc (t, Ti ) − ci /2 × 100 ×
j =1 Z(t, Tj ))
Z(t, Ti ) =
100 × (1 + ci /2)
47
(2.27)
This procedure is relatively simple to implement, as the example above shows. One
of the issues, though, is that bond data at six-month intervals are not always available.
Unfortunately, this procedure requires all of the bonds, because otherwise the iterative
procedure stops and there is no way to keep going. The appendix at the end of this chapter
reviews some other methodologies that are widely used to estimate the discount factors
Z(0, T ) from coupon bonds.
2.4.3
Expected Return and the Yield to Maturity
How can we measure the expected return on an investment in Treasury securities? Assuming the investor will hold the bond until maturity, computing the expected return on an
investment in a zero coupon bond is relatively straightforward, as the final payoff is known
and there are no intermediate cash flows. Thus, quite immediately, we have
Return on zero coupon bond =
1
−1
Z(t, T )
(2.28)
This is the return between t and T . It is customary to annualize this amount, so that
Annualized return on zero coupon bond =
1
Z(t, T )
T 1−t
−1
(2.29)
This, of course, corresponds to the annually compounded yield on the zero coupon, as in
Equation 2.6 for n = 1.
For coupon bonds it is more complicated. A popular measure of return on investment
for coupon bonds is called yield to maturity, which is defined as follows:
Definition 2.6 Let Pc (t, T ) be the price at time t of a Treasury bond with coupon c and
maturity T . Let Ti denote the coupon payments times, for i = 1, ..., n. The semi-annually
compounded yield to maturity, or internal rate of return, is defined as the constant rate
y that makes the discounted present value of the bond future cash flows equal to its price.
That is, y is defined by the equation
n
Pc (t, T ) =
i= 1
c/2 × 100
(1 + y/2)
2×(T i −t)
+
100
(1 + y/2)
2×(T n −t)
(2.30)
Before moving to interpret this measure of return on investment, it is important to
recognize a major distinction between the formula in Equation 2.30 and the one that we
obtained earlier in terms of discount factors, namely Equation 2.15. Although they appear
the same, it is crucial to note that the yield to maturity y is defined as the particular constant
rate that makes the right-hand side of Equation 2.30 equal to the price of the bond. Instead,
48
BASICS OF FIXED INCOME SECURITIES
Equation 2.15 is the one defining the price of the bond from the discount factors Z(t, T ).
Unless the term structure of interest rates is exactly flat, the yields at various maturities are
different, and will not coincide with the yield to maturity y. Indeed, to some extent, the
yield to maturity y can be considered an average of the semi-annually compounded spot
rates r2 (0, T ), which define the discount Z(0, T ). However, it is important to note that this
“average” depends on the coupon level c. In fact:
Fact 2.10 Two different bonds that have the same maturity but different coupon rates c
have different yield to maturities y.
This fact is easily illustrated with an example:
EXAMPLE 2.11
Columns 1 to 6 of Table 2.2 display coupon rates, maturities, and quotes of the
latest issued Treasury notes on February 15, 2008.8 Column 7 shows the discount
curve Z(0, T ) obtained from the bootstrap procedure discussed in Section 2.4.2, and
Column 8 reports the continuously compounded spot rate curve r(0, T ).
On February 15, 2008, traders could buy or sell two Treasury securities with the
same maturity T = 9.5 years, but with very different coupon rates. In particular, a Tnote with coupon c = 4.750% and a T-bond with coupon c = 8.875% were available.
Using the discount factors Z(0, T ) in Table 2.2 and the formula in Equation 2.15 we
can determine the fair prices of the two securities. In particular, we have9
Price T-note c= 4.750
=
=
Price T-bond c= 8.875
=
=
4.750
×
2
9.5
Z(0, T )
+ 100 × Z(0, 9.5)
T =0.5
107.8906
8.875
×
2
(2.31)
9.5
Z(0, T )
+ 100 × Z(0, 9.5)
T =0.5
141.5267
(2.32)
What are the yield to maturity of these two securities? Solving Equation 2.30 for the
two bonds, the yield to maturity of the c = 4.75 T-note and c = 8.875 T-bond are,
respectively
yc= 4.750
= 3.7548%.
(2.33)
yc= 8.875
= 3.6603%
(2.34)
As it can be seen from Equations 2.33 and 2.34, the bond with the higher coupon has
lower yield to maturity y.
8 Data
excerpted from CRSP (Daily Treasuries) ¤2009 Center for Research in Security Prices (CRSP), The
University of Chicago Booth School of Business.
9 We use fair prices, i.e., prices obtained from the same discount curve Z (0, T ), to better illustrate the concept
of yield to maturity and its relation to a bond coupon rate. It turns out however that on February 15, 2008, the
9.5-year T-bond with c = 8.875 was actually trading at 140.0781, about 1% less than its fair price computed in
Equation 2.32. This lower price is due to the lack of liquidity of bonds that have been issued long in the past
compared to the latest issued T-notes used to compute the discount curve Z (0, T ).
49
COUPON BONDS
Table 2.2 Term Structure on February 15, 2008
Coupon Maturity Time to
Rate (%)
Date
Maturity
4.125
4.500
4.875
4.750
4.125
5.000
5.000
4.875
4.375
3.875
4.250
4.000
4.250
4.000
4.250
4.500
4.875
4.625
4.750
3.500
8/15/2008
2/15/2009
8/15/2009
2/15/2010
8/15/2010
2/15/2011
8/15/2011
2/15/2012
8/15/2012
2/15/2013
8/15/2013
2/15/2014
8/15/2014
2/15/2015
8/15/2015
2/15/2016
8/15/2016
2/15/2017
8/15/2017
2/15/2018
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
Bootstrap Spot
Discount Curve
Bid
Ask
Mid
100.9844
102.6094
104.4766
105.5078
105.0859
108.2344
109.0000
109.1719
107.3281
105.1406
106.8125
105.2344
106.3281
104.3750
105.4063
106.7188
109.0000
106.9375
107.8750
97.8750
101.0156
102.6406
104.5078
105.5391
105.1172
108.2656
109.0313
109.2031
107.3594
105.1719
106.8438
105.2656
106.3594
104.4063
105.4375
106.7500
109.0313
106.9688
107.9063
97.9063
101.0000
102.6250
104.4922
105.5234
105.1016
108.2500
109.0156
109.1875
107.3438
105.1563
106.8281
105.2500
106.3438
104.3906
105.4219
106.7344
109.0156
106.9531
107.8906
97.8906
98.9590
98.1892
97.3147
96.2441
95.0822
93.7612
92.2213
90.6046
88.7259
86.9809
85.0858
83.1241
81.1114
79.0613
76.8759
74.8256
72.6763
70.8392
69.1582
68.1581
Discount with Discount with
y = 3.7548% y = 3.6603%
2.0930
1.8274
1.8147
1.9141
2.0172
2.1473
2.3137
2.4666
2.6582
2.7896
2.9365
3.0806
3.2207
3.3564
3.5064
3.6251
3.7548
3.8306
3.8818
3.8334
98.1572
96.3484
94.5729
92.8301
91.1194
89.4403
87.7920
86.1742
84.5862
83.0274
81.4974
79.9956
78.5214
77.0744
75.6541
74.2600
72.8915
71.5483
70.2298
68.9356
98.2027
96.4378
94.7045
93.0024
91.3309
89.6895
88.0775
86.4945
84.9400
83.4134
81.9142
80.4420
78.9962
77.5765
76.1822
74.8130
73.4684
72.1480
70.8513
69.5779
Data excerpted from CRSP (Daily Treasuries) ¤2009 Center for Research in Security Prices (CRSP), The University
of Chicago Booth School of Business.
To verify the yield to maturity computed are indeed correct, Table 2.2 also reports
the discount factors Z y (0, T ) = (1 + y/2)−2×T implied by Equations 2.33 and 2.34.
Using these two discounts Z y (0, T ) instead of Z(0, T ) in Equations 2.31 and 2.32,
respectively, we indeed obtain the correct prices:
9.5
Price T-note c=4.750
=
T = 0.5
4.750/2
1+
9.5
Price T-bond c=8.875
=
T = 0.5
3.7548%
2
2×T
+
2×T
+
8.875/2
1+
3.6603%
2
100
1+
3.7548%
2
2×9.5
= 107.8906
2×9.5
= 141.5267
100
1+
3.6603%
2
This example shows that there is something curious in the definition of yield to maturity.
Why does the coupon rate affect the yield to maturity? To understand the intuition, we
need to note that y correctly measures the expected return on an investment only under the
strict condition that the investor can reinvest all of the coupons at the rate y over the life of
the bond.
To see this, let us compute the total payoff at maturity T assuming that the investor
can reinvest all of the coupons paid at dates T1 , T2 , ..., Tn −1 at the constant rate y for the
remaining periods T − T1 , T − T2 , ..., T − Tn −1 . This is given by the following:
Total payoff at T
=
c × 100
2×(T −T 1 )
× (1 + y/2)
2
50
BASICS OF FIXED INCOME SECURITIES
+
...
c × 100
2×(T −T 2 )
× (1 + y/2)
2
c × 100
2×(T −T n −1 )
× (1 + y/2)
2
c × 100
+ 100
+
2
n
c × 100
2×(T −T i )
=
×
(1 + y/2)
2
i=1
+
+ 100
We can now compute the present value of the total payoff at T , using y as the constant
semi-annual yield and thus Z y (t, T ) = (1 + y/2)−2(T −t) as the discount factor. This gives
Present value of
total payoff at T
=
=
c × 100
×
Z (t, T ) ×
2
c × 100
×
2
=
c × 100
×
2
=
P (t, T )
n
i= 1
n
i= 1
n
(1 + y/2)
y
2×(T −T i )
+ 100
i=1
2×(T −T )
i
(1 + y/2)
2×(T
−t)
(1 + y/2)
1
(1 + y/2)2×(T i −t)
+
100
(1 + y/2)2×(T −t)
+
100
(1 + y/2)2×(T −t)
We find then that the price of the bond is (by definition) equal to the present value of the
total payoff at T , discounted at the yield to maturity y, under the assumption that all of the
coupons can be reinvested at same rate y over the life of the bond.
Given that it is practically impossible for an investor to reinvest all of the coupons at
the constant yield to maturity y , this latter measure is in fact a poor measure of expected
return. Indeed, the definition of a return on an investment cannot be given without a precise
definition of the time interval during which the security is held. For instance, a 10-year
STRIP provides the certain annualized return in Equation 2.29 if the security is held until
maturity. However, if the investor sells the bond after one year, the return may be higher or
lower than the promised yield (Equation 2.29) depending on what happens to interest rates.
A substantial increase in interest rates, for instance, will tend to lower prices of long-term
bonds, and thus the investor can end up with a capital loss. In later chapters we use modern
financial concepts to precisely define the expected return on an investment during a given
period, as well as the no arbitrage restrictions that must exist across bonds.
Why then do traders use the notion of yield to maturity y in their every day trading?
Given a coupon rate c, Equation 2.30 shows that there is a one-to-one relation between the
price Pc of the bond and the yield to maturity y. Thus, a trader can quote the same bond
by using Pc or y. To some extent, then, the yield to maturity is just a convenient way of
quoting a bond price to other traders.
2.4.4 Quoting Conventions
We end this section on Treasury bonds with a few remarks on the market quoting convention
for Treasury bills and Treasury bonds.
COUPON BONDS
51
2.4.4.1 Treasury Bills. Treasury bills are quoted on a discount basis. That is, rather
than quoting a price Pbill (t, T ) for a Treasury bill, Treasury dealers quote the following
quantity
100 − Pbill (t, T ) 360
×
(2.35)
d=
100
n
where n is the number of calendar days between t and T . For instance, on August 10, 2006
the Treasury issued a 182-day bill at a price of $97.477 for $100 of face value. Treasury
dealers quoted this price as
d=
100 − 97.477 360
×
= 4.99%
100
182
where d is an annualized discount rate on the face value of the Treasury bill.
Given a quote d from a Treasury dealer, we can compute the price of the Treasury bill
by solving for Pbill (t, T ) in Equation 2.35:
n
×d
(2.36)
Pbill (t, T ) = 100 × 1 −
360
2.4.4.2 Treasury Coupon Notes and Bonds. Coupon notes and bonds present
an additional complication. Between coupon dates, interest accrues on the bond. If a bond
is purchased between coupon dates, the buyer is only entitled to the portion of the coupon
that accrues between the purchase date and the next coupon date. The seller of the bond is
entitled to the portion of the coupon that accrued between the last coupon and the purchase
date. It is market convention to quote Treasury notes and bonds without any inclusion of
accrued interests. However, the buyer agrees to pay the seller any accrued interest between
the last coupon date and purchase price. That is, we have the formula
Invoice price = Quoted price + Accrued interest
(2.37)
The quoted price is sometimes referred to as the clean price while the invoice price is
sometimes also called dirty price.
The accrued interest is computed using the following intuitive formula:
Accrued interest
=
Interest due in the full period ×
×
Number of days since last coupon date
Number of days between coupon payments
Market conventions also determine the methodology to count days. There are three
main ways:
1. Actual/Actual: Simply count the number of calendar days between coupons;
2. 30/360: Assume there are 30 days in a month and 360 in a year;
3. Actual/360: Each month has the right number of days according to the calendar, but
there are only 360 days in a year.
Which convention is used depends on the security considered. For instance, Treasury bills use actual/360 while Treasury notes and bonds use the actual/actual counting
convention.
52
BASICS OF FIXED INCOME SECURITIES
2.5 FLOATING RATE BONDS
Floating rate bonds are coupon bonds whose coupons are tied to some reference interest rate.
The U.S. Treasury does not issue floating rate bonds, but governments of other countries as
well as individual corporations do. We present an example of floating rate bond in Example
2.12. It is important to spend some time on the pricing of floating rate bonds as a similar
methodology applies to numerous other interest rate securities, such as floaters and inverse
floaters (see Case Study in Section 2.8) as well as derivative instruments, such as interest
rate swaps (see Chapter 5).10
EXAMPLE 2.12
The Italian Treasury regularly issues CCT (Certificati di Credito del Tesoro), which
are floating rate bonds with 7 years to maturity. The CCT semi-annual coupon is
equal to the most recent rate on the six-month BOT (the Italian Treasury bill), plus a
spread (fixed at 0.15%). There is a six-month temporal lag between the determination
of the coupon and its actual payment.
Unless otherwise specified, we therefore define a floating rate bond as follows:
Definition 2.7 A semi-annual floating rate bond with maturity T is a bond whose coupon
payments at dates t = 0.5, 1, ..., T are determined by the formula
Coupon payment at t = c(t) = 100 × (r2 (t − 0.5) + s)
(2.38)
where r2 (t) is the 6-month Treasury rate at t, and s is a spread.11 Each coupon date is also
called reset date as it is the time when the new coupon is reset (according to the formula).
2.5.1 The Pricing of Floating Rate Bonds
The pricing of floating rate bonds is simple, although the logic may appear a little complicated at first. Consider the case in which the spread s = 0. In this case, the following is
true
Fact 2.11 If the spread s = 0, the ex-coupon price of a floating rate bond on any coupon
date is equal to the bond par value.12
To understand the logic, consider first the following simple example
10 In this chapter we only review the pricing of floating rate bonds for the case in which the coupon rate is linked to
the same interest rate that is also used for discounting purposes, which grealy simplifies the analysis and provides
the formulas needed for future applications.
11 For notational simplicity, in this section the six month rate is denoted by r (t) instead of r (t, t + 0.5).
2
2
12 Ex-coupon means that the price does not incorporate the coupon that is paid on that particular day. Par value is
the bond’s principal amount.
FLOATING RATE BONDS
53
EXAMPLE 2.13
Consider a one year, semi-annual floating rate bond. The coupon at time t = 0.5
depends on today’s interest rate r2 (0), which is known. If today r2 (0) = 2%, then
c(0.5) = 100 × 2%/2 = 1. What about the coupon c(1) at maturity T = 1? This
coupon will depend on the 6-month rate at time t = 0.5, which we do not know today.
This implies that we do not know the value of the final cash flow at time T = 1,
which is equal to 100 + c(1). Computing the present value of this uncertain cash
flow initially seems hard. And yet, with a moment’s reflection, it is actually simple.
Consider an investor who is evaluating this bond. This investor can project himself
to time t = 0.5, six months before maturity. Can the investor at time t = 0.5 guess
what the cash flow will be at time T = 1? Yes, because at time t = 0.5 the investor
will know the interest rate. So, he can compute what the value is at time t = 0.5.
Suppose that at time t = 0.5 the interest rate is r2 (0.5) = 3%, then the coupon at
time T = 1 is c(1) = 100 × r2 (0.5)/2 = 1.5. This implies that the value of the bond
at time t = 0.5 is equal to
Value bond at 0.5 = Present value of (100 + c(1)) =
100 + 1.5
= 100,
1 + 0.03/2
which is a round number, equal to par. What if the interest rate at time t = 0.5
was r2 (0.5) = 6%? In this case, the coupon rate at time T = 1 is c(1) = 100 ×
r2 (0.5)/2 = 103, and the value of the bond at t = 0.5 is
Value bond at 0.5 = Present value of (100 + c(1)) =
100 + 3
= 100
1 + 0.06/2
Still the same round number, equal to par. Indeed, independently of the level of
the interest rate r2 (T1 ), we find that the value of the bond at t = 0.5 is always equal
to 100:
Value bond at 0.5
=
=
=
Present value of (100 + c(1)) =
100 + 100 × r2 (0.5)/2
1 + r2 (0.5)/2
100 × (1 + r2 (0.5)/2)
1 + r2 (0.5)/2
100.
Even if the investor does not know the cash flow at time T = 1, because it depends
on the future floating rate r2 (0.5), the investor does know that at time t = 0.5 the
ex-coupon value of the floating rate bond will be 100, independently of what the
interest rate does then. But then, he can compute the value of the bond at time
t = 0, because the coupon at time T1 = 0.5 is known at time t = 0 as it is given by
c(0.5) = 100 × r2 (0)/2 = 101. Thus, the value at time t = 0 is
Value bond at 0
= Present value of (100 + c(0.5)) =
100 + 1
= 100
1 + .02/2
The result that at time t = 0.5 the ex-coupon bond price of this floating rate bond is always
equal to 100 may appear puzzling, but it is actually intuitive. When the interest rate moves
from r2 (0) = 2% to r2 (0.5) it has two effects:
1. It changes the future cash flow c(1) = 100 × r2 (0.5).
54
BASICS OF FIXED INCOME SECURITIES
• If the interest rate r2 (0.5) rises, the future cash flow increases.
• If the interest rate r2 (0.5) declines, the future cash flow declines.
2. It changes the discount rate to apply to the future cash flows.
• If the interest rate r2 (0, 5) rises, the discount rate increases.
• If the interest rate r2 (0, 5) declines, the discount rate declines.
The two effects, called “cash flow effect” and “discount effect,” work in opposite directions.
If the interest rate increases, the future cash flow increases, but it is discounted by a higher
rate.
The institutional feature of lagging the coupon payment by six months allows for this
cash flow and discount effect to exactly cancel each other out, leaving the value of the bond
at 100 at any reset date.
What if there are additional dates? The reasoning is the same, and we work backwards.
Table 2.3 contains the description of computations for the valuation of a 2-year floating rate
bond. Briefly, starting from the top of the table, the ex-coupon value at T = 2 is simply
the principal 100. The cum-coupon value is principal times the semi-annual interest rate
r2 (1.5)/2 determined six months earlier, at t = 1.5. We can compute the present value as
of t = 1.5 of the total cash flow at time T = 2, resulting in the ex-coupon price still equal
to 100. The logic is the same as in Example 2.13. The cum-coupon price at t = 1.5 is then
100 plus the coupon, which equals 100 times the semi-annual rate determined at t = 1.
Thus, the present value as of t = 1 of the total cash flow at time t = 1.5 (i.e., coupon plus
value of bond at t = 1.5) is equal to 100, again. And so on until t = 0.
2.5.2 Complications
We must discuss two simplifying assumptions made above: First, the spread s on the
floating rate is zero. Second, the time 0 of the valuation is a reset date. Fortunately, the
generalization to the more realistic case is simple.
First, if the spread s is nonzero we can decompose the total cash flow per period in two
parts, the floating part and the fixed part. This decomposition results in the equality
Floating coupon with spread s = Floating coupon with zero spread + Fixed coupon s
We can then value independently each component on the right-hand side, as we already
know how to value a floating coupon bond with zero spread (see previous section) and
a sequence of fixed coupon payments equal to s. Indeed, the present value of the fixed
T
sequence of payments equal to s is t= 0.5 s × Z(0, t). Therefore, we have the equality:
Price floating rate bond with spread s =
Price floating rate bond with zero spread
n
+s ×
Z(0, t)
t=0.5
Table 2.3
Time
Rate
Coupon
2
r2 (2)
c (2) = 100 ×
1.5
r2 (1.5)
The Valuation of a 2-year Floating Rate Bond
Ex-Coupon Price
r 2 (1 . 5 )
2
c (1.5) = 100 ×
r 2 (1 )
2
Cum-Coupon Price
c (2)
PFCR (2) = 100 + = 100 × 1 +
PF R (2) = 100
PF R (1.5)
=
=
P FC R (2 )
r (1.5)
1+ 2 2
r (1.5)
100× 1+ 2 2
= 100
1
r2 (1)
c (1) = 100 ×
r 2 (0 . 5 )
2
PF R (1)
=
=
0.5
r2 (0.5)
c (0.5) = 100 ×
r 2 (0 )
2
r (1)
1 + 22
r (1)
1 0 0 × 1 + 22
=
=
r2 (0)
-
=
PFCR (1.5) = PF R (1.5)
+ c (1.5)
= 100 × 1 + r 2 2(1 )
P FC R (1 )
r (0.5)
1+ 2 2
r (0.5)
100× 1+ 2 2
r (0.5)
1+ 2 2
P FC R (0 . 5 )
r (0)
1 + 22
r (0)
1 0 0 × 1 + 22
= 100
PFCR (1) = PF R (1) + c (1) = 100 × 1 + r 2 (02 . 5 )
r (0)
1 + 22
PFCR (0.5) = PF R (0.5)
+ rc2 (0.5)
= 100 × 1 + 2(0 )
FLOATING RATE BONDS
0
=
r (1)
1 + 22
= 100
PF R (0)
r (1.5)
1+ 2 2
P FC R (1 . 5 )
= 100
PF R (0.5)
r 2 (1 . 5 )
2
55
56
BASICS OF FIXED INCOME SECURITIES
At reset dates the price of the floating rate bond with zero spread is just par (=100), so that13
T
Price floating rate bond with spread s =
100 + s ×
Z(0, t)
(2.39)
t=0.5
The second complication is that the valuation may be outside reset dates. Consider first
Example 2.13. If today is not t = 0, but t = 0.25, how do we value the floating rate
bond? The backward induction argument up to t = 0.5 still holds: At time t = 0.5 the
ex-coupon bond price will be worth 100 and the cum-coupon bond price will be worth
$101 = $100 + $100 × 2%/2. The only difference from before is that we now have to
discount the amount 101 not back to t = 0 at the rate r2 (0) = 2%, but back to t = 0.25 at
the current 3-month rate. For instance, if the quarterly compounded 3-months rate is also
2%, that is r4 (0.25, 0.5) = 2%, then
Value bond at 0.25 = Present value of $101 =
$101
= $100.4975
(1 + 0.02/4)
In this case, the value of the bond depends on the current interest rate. If for instance
r4 (0.25, 0.5) = 1%, the value of the bond at 0.25 is $100.7481.
The same reasoning applies more generally. Let us denote by 0 the last reset date, and by
t the current trading day. Then, we know that at the next reset date, time 0.5, the ex-coupon
value of the floating rate bond will be $100. Thus, the cum-coupon value of the floating
rate bond at the next reset date is
PFCR (0.5, T ) = 100 + c(0.5) = 100 + 100 × r2 (0)/2
Therefore, the value at time 0 < t < 0.5 of this cash flow is
PF R (t, T ) = Present value of PFCR (0.5, T ) = Z(t, 0.5) × 100 × [1 + r2 (0)/2]
We summarize these results in the following:
Fact 2.12 Let T1 , T2 , ... Tn be the floating rate reset dates and let the current date t be
between time Ti and Ti+ 1 : Ti < t < Ti+ 1 . The general formula for a semi-annual floating
rate bond with zero spread s is
PF R (t, T ) = Z(t, Ti+ 1 ) × 100 × [1 + r2 (Ti )/2]
(2.40)
where Z(t, Ti+ 1 ) is the discount factor from t to Ti+1 . At reset dates, Z(Ti , Ti+1 ) =
1/(1 + r2 (Ti )/2), which implies
PF R (Ti , T ) = 100
(2.41)
It may be useful to note that although between coupon dates the value of a floating rate
bond depends on the interest rates, its sensitivity to variation in interest rate is very small,
as we shall see more fully in later chapters.
13 The spread s often reflects a lower credit quality than the reference rate used. The appropriate discount factors
should then be used to discount future cash flows. For instance, if the reference rate is the LIBOR, then the
LIBOR curve should be used, as discussed in Chapter 5.
SUMMARY
2.6
57
SUMMARY
In this chapter, we covered the following topics:
1. Discount factors: The discount factor is the value today of one dollar in the future.
Discount factors decrease with the time horizon and also vary over time.
2. Interest rates: The promised rate of return of an investment, an interest rate needs a
compounding frequency to be well defined. They are quoted on an annualized basis.
3. Compounding frequency: This is the frequency with which interest on an investment
is accrues over time. Continuous compounding refers to the limit in which payments
accrue every instant. Practically, daily compounding is very close to continuous
compounding.
4. Term structure of interest rates: The term structure of interest rate is the relation
between the interest rates and maturity. Investment horizons affect the interest rate
to be received on an investment or paid on a loan.
5. Zero coupon bonds: These are securities that pay only one given amount (par) at
maturity. Examples are Treasury bills or STRIPS.
6. Coupon notes and bonds. These are securities that pay a sequence of coupons and
the principal back at maturity. Examples are Treasury notes and bonds, which pay a
coupon semi-annually. T-notes have maturity up to ten years, while bonds have up
to thirty years.
7. Bootstrap: This procedure is for computing discount factors at various maturities
from data on coupon notes and bonds. It requires the availability of notes and bonds
at semi-annual intervals.
2.7
EXERCISES
1. Figure 2.3 shows that the term structure of interest rates can be declining, with
short-term spot rates higher than long-term spot rates. How steep can the decline
in spot rates be? Consider two STRIPS: One has 3-years to maturity and yields
a continuously compounded rate r(0, 3) = 10%, while the second has 5 years to
maturity and yields a continuously compounded rate r(0, 5) = 5%. Discuss whether
this scenario is possible, and, if not, what arbitrage strategy could be set up to gain
from the mispricing.
2. Compute the price, the yield and the continuously compounded yield for the following Treasury bills. For the 1-year Treasury bill also compute the semi-annually
compounded yield.
(a) 4-week with 3.48% discount (December 12, 2005)
(b) 4-week with 0.13% discount (November 6, 2008)
(c) 3-month with 4.93% discount (July 10, 2006)
(d) 3-month with 4.76% discount (May 8, 2007)
58
BASICS OF FIXED INCOME SECURITIES
(e) 3-month with 0.48% discount (November 4, 2008)
(f) 6-month with 4.72% discount (April 21, 2006)
(g) 6-month with 4.75% discount (June 6, 2007)
(h) 6-month with 0.89% discount (November 11, 2008)
(i) 1-year with 1.73% discount (September 30, 2008)
(j) 1-year with 1.19% discount (November 5, 2008)
3. You are given the following data on different rates with the same maturity (1.5 years),
but quoted on a different basis and different compounding frequencies:
• Continuously compounded rate: 2.00% annualized rate
• Continuously compounded return on maturity: 3.00%
• Annually compounded rate: 2.10% annualized rate
• Semi-annually compounded rate: 2.01% annualized rate
You want to find an arbitrage opportunity among these rates. Is there any one that
seems to be mispriced?
4. Using the semi-annually compounded yield curve in Table 2.4, price the following
securities:
(a) 5-year zero coupon bond
(b) 7-year coupon bond paying 15% semiannually
(c) 4-year coupon bond paying 7% quarterly
(d) 3 1/4-year coupon bond paying 9% semiannually
(e) 4-year floating rate bond with zero spread and semiannual payments
(f) 2 1/2-year floating rate bond with zero spread and annual payments
(g) 5 1/2-year floating rate bond with 35 basis point spread with quarterly payments
(h) 7 1/4-year floating rate bond with 40 basis point spread with semiannual payments
5. Consider a 10-year coupon bond paying 6% coupon rate.
(a) What is its price if its yield to maturity is 6%? What if it is 5% or 7%?
(b) Compute the price of the coupon bond for yields ranging between 1% and 15%.
Plot the resulting bond price versus the yield to maturity. What does the plot
look like?
6. Consider the data in Table 2.4. Consider two bonds, both with 7 years to maturity,
but with different coupon rates. Let the two coupon rates be 15% and 3%.
(a) Compute the prices and the yields to maturity of these coupon bonds.
(b) How do the yields to maturity compare to each other? If they are different, why
are they different? Would the difference in yields imply that one is a better
“buy” than the other?
59
EXERCISES
Table 2.4 Yield Curve on March 15, 2000
Maturity
Yield
Maturity
Yield
Maturity
Yield
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
6.33%
6.49%
6.62%
6.71%
6.79%
6.84%
6.87%
6.88%
6.89%
6.88%
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
6.86%
6.83%
6.80%
6.76%
6.72%
6.67%
6.62%
6.57%
6.51%
6.45%
5.25
5.50
5.75
6.00
6.25
6.50
6.75
7.00
7.25
7.50
6.39%
6.31%
6.24%
6.15%
6.05%
5.94%
5.81%
5.67%
5.50%
5.31%
Yields calculated based on data from CRSP (Daily Treasuries).
7. Today is May 15, 2000.
(a) Compute the discount curve Z(0, T ) for T =6 month, 1 year, 1.5 years, and 2
years from the following data:
• A 6-month zero coupon bond priced at $96.80 (issued 5/15/2000)
• A 1-year note with 5.75% coupon priced at $99.56 (issued 5/15/1998)
• A 1.5-year note with 7.5% coupon priced at $100.86 (issued 11/15/1991)
• A 2-year note with 7.5% coupon priced at $101.22 (issued 5/15/1992)
(b) Once you get the discount curve Z(0, T ) you take another look at the data and
you find the following 1-year bonds:
i. A 1-year note with 8.00% coupon priced at $101.13 (issued 5/15/1991)
ii. A 1-year bond with 13.13% coupon priced at $106.00 (issued 4/2/1981)
Compute the prices for these bonds with the discounts you found. Are the
prices the same as what the market says? Is there an arbitrage opportunity?
Why?
8. On May 15, 2000 you obtain the data on Treasuries in Table 2.5. Compute the
semiannual yield curve, spanning over 9 years, from the data using the bootstrap
procedure.
9. The Orange County case study at the end of the chapter discusses the pricing of inverse
floaters, and provides a decomposition of inverse floaters in terms of a coupon bond,
a floating rate bond, and a zero coupon bond (see Equation 2.43). Find an alternative
decomposition of the same security, and compute the price. Do you obtain the same
price? Discuss your findings in light of the law of one price discussed in Chapter 1.
60
BASICS OF FIXED INCOME SECURITIES
Table 2.5
Bonds and Notes on May 15, 2000
Cusip
Issue Date
Maturity Date
Name
Coupon
Bid
Ask
912827ZE
912827ZN
912810CT
912810CU
912810CW
912810CX
912810CZ
912827F4
912827G5
912810DA
912810DC
912810DD
912810DE
912810DG
912827N8
912810DH
912810DK
912810DM
912827S8
912810DQ
912810DR
912827V8
912810DU
912827X8
912827Y5
912827Z6
9128272J
9128272U
9128273E
9128273X
9128274F
9128274V
9128275G
8/15/1990
11/15/1990
1/12/1981
4/2/1981
7/2/1981
10/7/1981
1/6/1982
5/15/1992
8/15/1992
9/29/1982
1/4/1983
4/4/1983
7/5/1983
10/5/1983
2/15/1994
4/5/1984
7/10/1984
10/30/1984
2/15/1995
4/2/1985
7/2/1985
11/15/1995
1/15/1986
5/15/1996
7/15/1996
10/15/1996
2/15/1997
5/15/1997
8/15/1997
2/15/1998
5/15/1998
11/16/1998
5/17/1999
8/15/2000
11/15/2000
2/15/2001
5/15/2001
8/15/2001
11/15/2001
2/15/2002
5/15/2002
8/15/2002
11/15/2002
2/15/2003
5/15/2003
8/15/2003
11/15/2003
2/15/2004
5/15/2004
8/15/2004
11/15/2004
2/15/2005
5/15/2005
8/15/2005
11/15/2005
2/15/2006
5/15/2006
8/15/2006
11/15/2006
2/15/2007
5/15/2007
8/15/2007
11/15/2007
5/15/2008
11/15/2008
5/15/2009
NOTE
NOTE
BOND
BOND
BOND
BOND
BOND
NOTE
NOTE
BOND
BOND
BOND
BOND
BOND
NOTE
BOND
BOND
BOND
NOTE
BOND
BOND
NOTE
BOND
NOTE
NOTE
NOTE
NOTE
NOTE
NOTE
NOTE
NOTE
NOTE
NOTE
8.750%
8.500%
11.750%
13.125%
13.375%
15.750%
14.250%
7.500%
6.375%
11.625%
10.750%
10.750%
11.125%
11.875%
5.875%
12.375%
13.750%
11.625%
7.500%
12.000%
10.750%
5.875%
9.375%
6.875%
7.000%
6.500%
6.250%
6.625%
6.125%
5.500%
5.625%
4.750%
5.500%
100.5742
100.8906
103.8047
105.9805
107.6406
112.3945
111.9297
101.2031
99.0469
110.6680
109.5117
110.3281
112.1523
115.3086
97.1172
118.8984
124.9375
118.2969
102.8633
121.6133
117.0664
95.9844
112.0352
100.6055
101.2031
98.7500
97.4883
99.5625
96.7578
93.1328
93.7852
87.9766
92.8242
100.6055
100.9219
103.8359
106.0117
107.6719
112.4258
111.9609
101.2344
99.0781
110.6992
109.5430
110.3594
112.1836
115.3398
97.1484
118.9297
124.9688
118.3281
102.8945
121.6445
117.0977
96.0156
112.0664
100.6367
101.2344
98.7813
97.5195
99.5938
96.7891
93.1641
93.8164
88.0078
92.8555
Data excerpted from CRSP (Daily Treasuries) ¤2009 Center for Research in Security Prices (CRSP),
The University of Chicago Booth School of Business.
61
CASE STUDY: ORANGE COUNTY INVERSE FLOATERS
2.8
CASE STUDY: ORANGE COUNTY INVERSE FLOATERS
With the tools we have developed in this chapter we can price all sorts of securities.14 An
interesting security to price is an inverse floater. These securities became very popular
during the Orange County bankruptcy at the end of 1994, since it is estimated that a
significant fraction of the county’s portfolio comprised these securities. The bankruptcy of
Orange County is a classic example of the risk that is inherent in interest rate securities, and
we will discuss this case study more thoroughly in Chapters 3 and 4 after we introduce some
tools to measure interest rate risk. In this section we pave the way for the discussion of risk
in the next chapters by investigating the methodology to value inverse floaters. Because we
are interested in understanding the dynamics behind Orange County’s financial problems,
we assume that we are beginning our analysis on December 31, 1993 (a year before the
county declared bankruptcy).
2.8.1
Decomposing Inverse Floaters into a Portfolio of Basic Securities
An inverse floater is a security that pays a lower coupon as interest rates go up (hence the
name inverse floater). For this to work, we need to establish a fixed reference rate from
which to subtract the floating rate. To keep things simple, we assume that the inverse floater
promises to pay 15% minus the short-term interest rate on an annual basis with 3 years
maturity. That is, the coupon on the bond is:
c(t) = 15% − r1 (t − 1)
(2.42)
where r1 (t − 1) denotes the annually compounded rate at time t − 1, and we adopt the
usual convention according to which the cash flow at time t, c(t), depends on the interest
rate one period earlier, namely t − 1 in our case as payments are annual. The assumption
of the annual payment frequency for the inverse floater is made for simplicity, so that
the calculations are easier to follow. Notice also that Equation 2.42 contains a further
simplification, namely, the fact that the coupon is always positive, which would be violated
if the short rate were ever larger than 15%. In reality, if this situation occurs, the bond does
not pay any coupon [i.e., c(t) = 0 if r(t − 1) > 15%]. To take this case into account we
need to develop further tools, as we will do in Parts II and III of this book. For the time
being assume that we know with certainty that the short rate is always below 15%.
An interesting fact arises by looking at the formula in Equation 2.42: Coupon payments
are a combination of a fixed rate and a floating rate bond. So, for the coupon payments,
this is the same as having a long position in a fixed rate bond and a short position in a
floating rate bond, as such positions would entail receiving a fixed coupon and paying a
floating coupon.15 If we follow this strategy, however, we find that at maturity T = 3 the
principal we receive from the fixed rate bond has to be used to pay for the principal of the
floating rate bond. That is, only a long position in the fixed rate and short position in the
floating rate bond does not exactly mimic an inverse floater. We can solve this problem by
14 Thanks to Francisco Javier Madrid for his help in putting this case together.
Descriptive material is from the case
study ERISK: Orange County, downloaded from http://www.erisk.com/Learning/CaseStudies/OrangeCounty.asp.
15 An investor has a long position in a bond if he holds the bond in the portfolio. In contrast, a short position means
that the investor sold the bond without actually having it in the portfolio. The short position is accomplished by
first borrowing the bond from a broker, typically in the repo market, and then selling it to the market. It is the
responsibility of the investor who sold the bond short to make the regular coupon payments to the counterparty.
62
BASICS OF FIXED INCOME SECURITIES
adding to the portfolio a 3-year zero coupon bond. From the law of one price (see Fact 1.1
in Chapter 1) the price of an inverse floater is then:
Price inverse floater = Pz (0, 3) + Pc (0, 3) − PF R (0, 3)
(2.43)
where we recall that Pz (0, 3), Pc (0, 3), and PF R (0, 3) denote the prices of a zero coupon
bond, a coupon bond, and a floating rate bond with three years to maturity.
2.8.2 Calculating the Term Structure of Interest Rates from Coupon Bonds
The next challenge is to determine the term structure of the interest rates so as to obtain
the discount rates for the bonds. A first idea might be to find zero coupon bonds for all
these periods. The problem is that we might not necessarily find all the data we want. An
alternative is to look at all bonds that are being quoted in the market today (December 31,
1993) and use this data to plot the yield curve. The reasoning is that every day, quotes
are available on bonds maturing at different dates. As we saw in this chapter, in absence
of arbitrage opportunities, any bond with coupon c that matures in three years (even if it
was issued, for example, seven years ago), must have the same price as a 3-year bond,
issued today, with coupon c. Additionally, through Equations 2.26 and 2.27 we can convert
coupon paying bonds into zero coupon bonds. This exercise, however, may prove more
challenging than it sounds. Look at Table 2.6, which reports all the bond price quotes
available on December 31, 1993. There are 224 bonds quoted. To perform the bootstrap
analysis, we need many fewer bonds. How do we pick the bonds to bootstrap out the term
structure of interest rates?
After some careful (and time consuming) analysis of the data in Table 2.6, we resolve
to use the subsample of data, contained in Table 2.7, which are nicely spaced at 6-month
intervals. The last two columns of the table provide the discount factors Z(0, T ) computed
from either bid prices or ask prices.16
2.8.3 Calculating the Price of the Inverse Floater
Recall that the value of the inverse floater can be computed from the value of a zero
coupon bond, a coupon bond with coupon rate equal to 15%, and a floating rate bond, all
of them with maturity equal to three years. Given the discount factors in Table 2.7, we
can obtain values for these standard bonds. For simplicity, we use the average discount
Z(0, T ) = 0.5 × ZA (0, T ) + 0.5 × ZB (0, T ) for the following calculations.
1. Three-year zero coupon bond. Three years from December 31, 1993 corresponds
to the maturity date December 31, 1996. The discount factor Z(0, 3) = 0.8745.
Thus, Pz (0, 3) = 100 × 0.8745 = $87.45.
2. Three-year, 15% fixed coupon bond. Given the discount factors Z(0, T ) for
T = 1, 2, 3, we can compute the price of a coupon bond by applying the bond
pricing formula in Equation 2.13, with the only caveat that in this exercise coupons
are annually paid, and thus we do not have to divide them by 2 (as instead we do in
Equation 2.13). More precisely, Table 2.8 carries out the calculation, and obtains the
price of the fixed coupon bond Pc (0, T ) = $128.83.
16 Recall
that the bid and ask prices are the quotes at which security dealers are ready to buy or sell the securities.
Because they make a profit from the spread between them, the ask price is higher than the bid price.
CASE STUDY: ORANGE COUNTY INVERSE FLOATERS
Table 2.6 Bond Quotes on December 31, 1993
Maturity
Coupon
Bid
Ask
Maturity
Coupon
Bid
Ask
Maturity
Coupon
Bid
Ask
19940106
19940113
19940115
19940120
19940127
19940131
19940203
19940210
19940215
19940215
19940215
19940217
19940224
19940228
19940303
19940310
19940317
19940324
19940331
19940331
19940331
19940407
19940414
19940415
19940421
19940428
19940430
19940505
19940512
19940515
19940515
19940515
19940519
19940526
19940531
19940602
19940609
19940616
19940623
19940630
19940630
19940630
19940715
19940728
19940731
19940815
19940815
19940815
19940815
19940825
19940831
19940922
19940930
19940930
19941015
19941020
19941031
19941115
19941115
19941115
19941115
19941117
19941130
19941215
19941231
19941231
19950115
19950131
19950215
19950215
19950215
19950215
19950215
19950228
19950331
0.000
0.000
7.000
0.000
0.000
4.875
0.000
0.000
9.000
6.875
8.875
0.000
0.000
5.375
0.000
0.000
0.000
0.000
5.750
8.500
0.000
0.000
0.000
7.000
0.000
0.000
5.375
0.000
0.000
7.000
9.500
13.125
0.000
0.000
5.125
0.000
0.000
0.000
0.000
5.000
8.500
0.000
8.000
0.000
4.250
8.750
6.875
8.625
12.625
0.000
4.250
0.000
4.000
8.500
9.500
0.000
4.250
10.125
6.000
8.250
11.625
0.000
4.625
0.000
4.625
7.625
8.625
4.250
3.000
10.500
5.500
7.750
11.250
3.875
3.875
99.960
99.904
100.094
99.841
99.788
100.125
99.724
99.666
100.688
100.406
100.656
99.612
99.551
100.344
99.489
99.423
99.367
99.306
100.625
101.250
99.248
99.173
99.113
101.031
99.047
98.990
100.656
98.920
98.849
101.313
102.219
103.563
98.780
98.714
100.719
98.653
98.582
98.516
98.439
100.813
102.500
98.391
102.406
98.113
100.469
103.188
102.063
103.125
105.531
97.841
100.469
97.563
100.313
103.594
104.438
97.249
100.531
105.438
101.969
103.875
106.750
96.924
100.875
96.646
100.906
103.813
104.875
100.531
100.250
107.344
101.844
104.281
108.125
100.063
100.031
99.961
99.908
100.156
99.846
99.795
100.188
99.728
99.671
100.750
100.469
100.719
99.617
99.557
100.406
99.492
99.427
99.371
99.311
100.688
101.313
99.253
99.178
99.119
101.094
99.053
98.997
100.719
98.927
98.856
101.375
102.281
103.625
98.788
98.723
100.781
98.661
98.591
98.525
98.449
100.875
102.563
98.401
102.469
98.125
100.531
103.250
102.125
103.188
105.594
97.854
100.531
97.578
100.375
103.656
104.500
97.265
100.594
105.500
102.031
103.938
106.813
96.942
100.938
96.665
100.969
103.875
104.938
100.594
101.250
107.406
101.906
104.344
108.188
100.125
100.094
19950415
19950430
19950515
19950515
19950515
19950515
19950515
19950531
19950630
19950715
19950731
19950815
19950815
19950815
19950831
19950930
19951015
19951031
19951115
19951115
19951115
19951115
19951130
19951231
19960115
19960131
19960215
19960215
19960215
19960229
19960331
19960415
19960430
19960515
19960515
19960531
19960630
19960715
19960731
19960815
19960831
19960930
19961015
19961031
19961115
19961115
19961130
19961231
19970115
19970131
19970228
19970331
19970415
19970430
19970515
19970531
19970630
19970715
19970731
19970815
19970831
19970930
19971015
19971031
19971115
19971130
19971231
19980115
19980131
19980215
19980228
19980331
19980415
19980430
19980515
8.375
3.875
10.375
12.625
5.875
8.500
11.250
4.125
4.125
8.875
4.250
4.625
8.500
10.500
3.875
3.875
8.625
3.875
11.500
5.125
8.500
9.500
4.250
4.250
9.250
7.500
4.625
7.875
8.875
7.500
7.750
9.375
7.625
4.250
7.375
7.625
7.875
7.875
7.875
4.375
7.250
7.000
8.000
6.875
4.375
7.250
6.500
6.125
8.000
6.250
6.750
6.875
8.500
6.875
8.500
6.750
6.375
8.500
5.500
8.625
5.625
5.500
8.750
5.750
8.875
6.000
6.000
7.875
5.625
8.125
5.125
5.125
7.875
5.125
9.000
105.500
99.938
108.500
111.563
102.531
106.000
109.656
100.250
100.188
107.125
100.281
100.875
106.875
110.031
99.688
99.594
107.625
99.531
113.063
101.688
107.688
109.531
100.063
100.000
109.625
106.344
100.688
107.125
109.156
106.500
107.188
110.750
107.094
99.719
106.625
107.375
108.688
108.125
108.219
99.844
106.813
106.594
108.938
106.125
99.563
107.156
105.250
104.531
109.469
104.594
106.063
106.531
111.375
106.563
111.531
106.219
105.156
111.875
102.344
112.469
102.563
102.188
113.125
102.969
113.781
103.781
103.781
110.438
102.281
111.500
100.438
100.344
110.719
100.250
115.250
105.563
100.000
108.563
111.688
102.594
106.063
109.719
100.313
100.250
107.188
100.344
100.938
106.938
110.094
99.750
99.656
107.688
99.594
113.188
101.750
107.750
109.594
100.125
100.063
109.688
106.406
100.750
107.188
109.219
106.563
107.250
110.813
107.156
99.781
106.688
107.438
108.750
108.188
108.281
99.906
106.875
106.656
109.000
106.188
99.625
107.219
105.313
104.594
109.531
104.656
106.125
106.594
111.438
106.625
111.594
106.281
105.219
111.938
102.406
112.531
102.625
102.250
113.188
103.031
113.844
103.844
103.844
110.500
102.344
111.563
100.500
100.406
110.781
100.313
115.313
19980531
19980630
19980715
19980731
19980815
19980831
19980930
19981015
19981031
19981115
19981115
19981130
19981231
19990115
19990215
19990415
19990515
19990715
19990815
19991015
19991115
20000115
20000215
20000415
20000515
20000815
20001115
20010215
20010215
20010515
20010515
20010815
20010815
20011115
20011115
20020215
20020515
20020815
20021115
20030215
20030215
20030515
20030815
20030815
20031115
20040515
20040815
20041115
20050515
20050815
20060215
20150215
20150815
20151115
20160215
20160515
20161115
20170515
20170815
20180515
20181115
20190215
20190815
20200215
20200515
20200815
20210215
20210515
20210815
20211115
20220815
20221115
20230215
20230815
5.375
5.125
8.250
5.250
9.250
4.750
4.750
7.125
4.750
3.500
8.875
5.125
5.125
6.375
8.875
7.000
9.125
6.375
8.000
6.000
7.875
6.375
8.500
5.500
8.875
8.750
8.500
11.750
7.750
13.125
8.000
13.375
7.875
15.750
7.500
14.250
7.500
6.375
11.625
10.750
6.250
10.750
11.125
5.750
11.875
12.375
13.750
11.625
12.000
10.750
9.375
11.250
10.625
9.875
9.250
7.250
7.500
8.750
8.875
9.125
9.000
8.875
8.125
8.500
8.750
8.750
7.875
8.125
8.125
8.000
7.250
7.625
7.125
6.250
101.125
100.063
112.500
100.469
116.750
98.375
98.375
108.188
98.188
98.969
115.750
99.625
99.625
105.031
116.156
107.844
117.875
105.000
112.938
103.219
112.625
105.063
116.125
100.844
118.375
118.000
116.844
136.219
113.719
145.313
114.344
147.875
113.813
164.031
111.563
155.656
111.844
104.250
140.344
134.813
103.250
135.313
138.656
99.625
144.750
149.969
161.406
145.219
149.375
139.531
128.906
154.250
147.375
138.750
131.531
108.188
111.031
125.938
127.594
130.969
129.625
128.219
119.094
123.906
127.219
127.281
116.438
119.719
119.719
118.406
109.094
114.156
108.156
98.656
101.188
100.125
112.563
100.531
116.813
98.438
98.438
108.250
98.250
99.969
115.813
99.688
99.688
105.094
116.219
107.906
117.938
105.063
113.000
103.281
112.688
105.125
116.188
100.906
118.438
118.063
116.906
136.344
113.781
145.438
114.406
148.000
113.875
164.156
111.625
155.781
111.906
104.313
140.469
134.938
103.313
135.438
138.781
99.688
144.875
150.094
161.531
145.344
149.500
139.656
129.031
154.313
147.438
138.813
131.594
108.250
111.094
126.000
127.656
131.031
129.688
128.281
119.156
123.969
127.281
127.344
116.500
119.781
119.781
118.469
109.156
114.219
108.219
98.719
Data excerpted from CRSP (Daily Treasuries) ¤2009 Center for Research in Security Prices (CRSP), The University of Chicago Booth School of Business.
63
64
BASICS OF FIXED INCOME SECURITIES
Table 2.7
Discount Factors Z(0, T ) on December 31, 1993
Maturity
Coupon
Bid
Ask
ZB (0, T )
ZA (0, T )
19940630
19941231
19950630
19951231
19960630
19961231
19970630
19971231
19980630
19981231
0.000
7.625
4.125
4.250
7.875
6.125
6.375
6.000
5.125
5.125
98.3911
103.8125
100.1875
100.0000
108.6875
104.5313
105.1563
103.7813
100.0625
99.6250
98.4012
103.8750
100.2500
100.0625
108.7500
104.5938
105.2188
103.8438
100.1250
99.6875
0.9839
0.9639
0.9423
0.9191
0.9014
0.8743
0.8466
0.8203
0.7944
0.7703
0.9840
0.9645
0.9429
0.9196
0.9019
0.8748
0.8471
0.8208
0.7950
0.7708
Data Source: CRSP.
Table 2.8 The Price of a 15% Fixed Coupon Bond
Date
Cash Flow
Discount
Z(0, T )
Discounted
Cash Flow
19931231
19941231
19951231
19961231
0.15
0.15
1.15
0.9642
0.9193
0.8745
0.1446
0.1379
1.0057
Sum
Price (×100)
1.2883
128.83
3. Three-year floating rate bond. From Section 2.5, we recall that the value of a
floating rate bond is always equal to par at reset dates. Thus, we have PF R (3) = $100.
In conclusion, the value of the inverse floater is given by
Price inverse floater
= Pz (0, 3) + Pc (0, 3) − PF R (0, 3)
(2.44)
=
$87.45 + $128.83 − $100
(2.45)
=
$116.28
(2.46)
2.8.4 Leveraged Inverse Floaters
Within Orange County’s portfolio there were many different types of inverse floaters (e.g.
different maturities and maximum interest rates). In addition, the portfolio contained some
leveraged inverse floaters.17 The main difference between these and the plain vanilla inverse
17 See Mark Grinblatt and Sheridan Titman, Financial Markets and Corporate Strategy (2nd Edition), McGraw-Hill
Primis, 2006, Chapter 23.
APPENDIX: EXTRACTING THE DISCOUNT FACTORS Z (0, T ) FROM COUPON BONDS
65
floaters discussed earlier is that the parity of floating rate to fixed rate is greater than one.
For example, consider a 3-year leveraged inverse floater that pays a coupon of 25% minus
two times the short-term interest rate. To price this security, we need to revise the steps we
took to price inverse floaters.18 The coupon is given by
c(t) = 25% − 2 × r1 (t − 1)
(2.47)
What is a portfolio of bonds that pays this cash flow? A portfolio that is long a 25% fixed
coupon bond and short two floating rate bonds achieves the coupon described in Equation
2.47. However, such a position at maturity entails that we receive $100 from the long
position and we must pay $200 from the short position. In order to receive $100 overall,
we must also be long two zero coupon bonds. Thus, overall, we have
Price leveraged inverse floater = 2 × Pz (0, 3) + Pc (0, 3) − 2 × PF R (0, 3)
(2.48)
We already know from the previous section the prices Pz (0, 3) = $87.45 and PF R (0, 3) =
$100. The computation of the price of fixed-coupon bond with coupon rate equal to 25%
yields a price of Pc (0, 3) = $156.41. Thus, we immediately find
Price leveraged inverse floater
=
2 × $87.45 + $156.41 − 2 × $100
(2.49)
=
$131.32
(2.50)
This case study illustrates that we can readily apply the tools covered in this chapter
to value more complex securities, such as inverse floaters and leveraged inverse floaters.
In the next chapter we will follow up with this analysis and study the risk embedded
in these securities. Finally, we note that dealing with real data and real markets often
poses additional problems in the valuation and risk analysis of fixed income instruments:
For instance, the computation of the discount curve Z(0, t) requires the analysis of the
data in Table 2.6, which is not straightforward. The next section illustrates additional
methodologies used in practice to deal with such large quantities of data.
2.9
APPENDIX: EXTRACTING THE DISCOUNT FACTORS Z(0, T ) FROM
COUPON BONDS
The Orange County case study in the previous section makes it apparent that the bootstrap
methodology discussed in Section 2.4.2 has limited applicability, and this for two reasons.
First, for short-term maturities, there are too many bonds that mature on the same day
to choose from. To perform the bootstrap methodology, we then must cherry pick the
bonds that we deem have the highest liquidity (e.g., notes over bonds). Second, for longer
maturities not all of the bonds may be available. In this case, some approximation is
necessary. Sometimes it is possible to use the bonds that expire a few days earlier or later
than the ones in the six-month cycle needed for the bootstrap. But often the gap across
maturities may span longer periods, in which case the bootstrap methodology does not
work well.
18 We
maintain the assumption that the coupon is always positive, that is that rates are always below 25%/2
66
BASICS OF FIXED INCOME SECURITIES
2.9.1 Bootstrap Again
The iterative procedure described in the text is simple, but cumbersome. An easier way to
obtain the same result is to use the matrix notation. Let t = 0, for convenience, so that T
denotes both maturity date and time to maturity. Every coupon bond i is characterized by a
series of cash flows and a maturity T i . We can denote the total cash flow paid at time Tj as
ci (Tj ). In particular, denoting ci the coupon rate of bond i, we have ci (Tj ) = 100 × ci /2
for Tj < T i and ci (T i ) = 100 × (1 + ci /2) and finally ci (Tj ) = 0 for Tj > T i . We can
put these cash flows in a row vector as follows:
Ci = ci (T1 ) , ci (T2 ) , ..., ci (Tn )
We can denote by Z (0) the vector of discount factors for various maturities Ti , that is
⎛
⎜
⎜
Z (0) = ⎜
⎝
Z (0, T1 )
Z (0, T2 )
..
.
⎞
⎟
⎟
⎟
⎠
Z (0, Tn )
The price of a coupon bond can be written using vector multiplication as:
Pci (0, T ) = Ci ×Z (0)
We can denote the column vector of bond prices available at time 0 as
⎛
⎜
⎜
P (0) = ⎜
⎝
Pc (0, T1 )
Pc (0, T2 )
..
.
⎞
⎟
⎟
⎟
⎠
Pc (0, Tn )
We then obtain a system of n equations with n unknowns [the unknowns are the values of
Z (0, T1 ) , ..., Z (0, Tn )]
P (0) = C × Z (0)
where C is the cash flow matrix:
⎛
⎜
⎜
C= ⎜
⎝
c1 (T1 )
c2 (T1 )
..
.
c1 (T2 )
c2 (T2 )
...
...
..
.
c1 (Tn )
c2 (Tn )
..
.
cn (T1 ) cn (T2 ) ...
cn (Tn )
⎞
⎟
⎟
⎟
⎠
Essentially, each row i of C corresponds to the cash flows of bond i for all the maturities
T1 ,...,Tn . In contrast, each column j describes all the cash flows that occur on that particular
maturity Tj from the n bonds. The discount factors can then be obtained by inverting the
cash flow matrix:
Z (0) = C−1 × P (0)
APPENDIX: EXTRACTING THE DISCOUNT FACTORS Z (0, T ) FROM COUPON BONDS
2.9.2
67
Regressions
As mentioned, we rarely have such nicely spaced data. Sometimes we in fact have too
many maturities and sometimes we do not have enough maturities available to carry out
the bootstrap procedure. The regression methodology deals with the case in which there
are too many bonds compared to the number of maturities. This is typically the case when
we consider maturities up to five years. For instance, in Table 2.6 there are 164 bonds with
maturity of less than five years, but there are only 60 months in five years, implying that
many months have multiple bonds maturing in them.
When we compute the cash flow matrix:
⎞
⎛ 1
c (T1 ) c1 (T2 ) ... c1 (Tn )
⎜ c2 (T1 ) c2 (T2 ) ... c2 (Tn ) ⎟
⎟
⎜
C =⎜ .
⎟
. . ..
⎠
⎝ ..
. .
cN (T1 )
cn (T2 ) ...
cn (Tn )
we end up with N rows (N = number of bonds) and n < N columns (n = number of
maturities). Since the solution to bootstrap involves inverting the matrix C, we have a
problem, as the matrix C must have an equal number of rows and columns to be inverted.
We can slightly change the bootstrap methodology to deal with this problem. For every
bond i = 1, ..., N let
(2.51)
Pci 0, T i = Ci ×Z (0) + εi
where εi is a random error term that captures any factor that generates the “mispricing.”
These factors include data staleness, lack of trading or liquidity. If we look at Equation
2.51, we see a regression equation of the type
n
β j xij + εj
yi = α +
j=1
where the data are y i = Pci (0, T i ) and xij = Cij , and the regressors are β j = Z (0, Tj ).
From basic Ordinary Least Squares (OLS) formulas, we then find
Z (0) = (C × C)
−1
C × P (0)
For this procedure to work, however, we must have more bonds than maturities, which does
not occur for longer maturities. Curve fitting treats this latter problem.
2.9.3
Curve Fitting
Let’s consider approaching the problem from a completely different angle. In particular,
we can postulate a parametric functional form for the discount factor Z(0, T ) as a function
of maturity T and use the current bond prices to estimate the parameters of this functional
form. One popular model is the following:
2.9.3.1 The Nelson Siegel Model The Nelson Siegel model is perhaps the most
famous model. The discount factor is posited to be given by
Z(0, T ) = e−r (0,T )T
(2.52)
68
BASICS OF FIXED INCOME SECURITIES
where the continously compounded yield with maturity T is given by
1 − e− λ
T
r(0, T ) = θ0 + (θ1 + θ2 )
T
λ
− θ2 e− λ
T
(2.53)
where θ0 , θ1 , θ2 and λ are parameters to be estimated from the current bond data.
The estimation proceeds as follows. For given parameter values (θ0 , θ1 , θ2 , λ), it is
possible to compute the value of bond prices implied by the Nelson Siegel model. For each
bond i = 1, .., N with coupon ci and cash flow payment dates maturity Tji , for j = 1, .., ni ,
the Nelson Siegel model implies that the bond price should be
⎛
⎞
i ni
c
Z(0, Tji ) + Z(0, T i )⎠
(2.54)
Pci, N S m odel = 100 × ⎝
2 j =1
For the same bond, we have the price quoted in the market, Pci, data (note that this has to be
the invoice price and not the quoted price). For each given set of parameters (θ0 , θ1 , θ2 , λ)
we can compute the difference between model prices and data. Namely, we can compute
N
J(θ0 , θ1 , θ2 , λ) =
Pci, N S
m odel
− Pci, data
2
(2.55)
i= 1
The Nelson Siegel model works perfectly if the model prices equal the data, i.e., if for
every i = 1, ..., N we have Pci, N S m odel = Pci, data . In this case, J(θ0 , θ1 , θ2 , λ) = 0.
The set of parameters (θ0 , θ1 , θ2 , λ) that achieves this objective would be the one to use to
determine the discount factors Z(0, T ).
In general, however, it will not be possible to find parameter values that price all of
the bonds exactly, because of staleness in the data, lack of liquidity, or lack of degrees
of freedom in the Nelson Siegel model (we only have four parameters, after all). Therefore, the procedure is instead to find parameters (θ0 , θ1 , θ2 , λ) that minimize the quantity
J(θ0 , θ1 , θ2 , λ) in Equation 2.55.
Figure 2.6 compares three methodologies of computing the term structure of interest
rates: The bootstrap, the Nelson Siegel model, and the Extended Nelson Siegel model,
further discussed below. The data are those contained in Table 2.6. The results of the
bootstrap methodology are already contained in Table 2.7 in the form of discount function
Z(0, T ). As it can be seen, the bootstrap method generates a yield curve that has a dip at
maturity T = 2.5. It is not clear from the data why the dip in yield is present at that point:
It could be a liquidity issue, or staleness, or simply an error in the database. The problem
with bootstrap is that correcting for these sources of imprecision is hard.
The solid line in Figure 2.6 plots the fitted yield curve according to the Nelson Siegel
model. The parameter estimates are θ 0 = 0.0754, θ1 = −0.0453, θ2 = −7.3182×10−009
and λ = 3.2286. The Nelson Siegel curve cuts through the bootstrapped curve smoothly.
If the dip of the 2.5 year yield was a data error, it gets corrected in the minimization of
errors. Indeed, note that we did not use only the ten bonds in Table 2.7 to fit the Nelson
Siegel model, but the whole of 161 bonds with maturity less than five years in Table 2.6.
Can the Nelson Siegel model fit all of these data reasonably well? Figure 2.7 plots both
the bond prices (stars) and the model prices (diamonds) for the various maturities: The
model works well if the stars are close to the diamonds. The figure shows that indeed for
APPENDIX: EXTRACTING THE DISCOUNT FACTORS Z (0, T ) FROM COUPON BONDS
69
most bonds this is the case, indicating that the model is doing quite well. The figure also
shows that indeed at T = 2.5 there is a star that differs substantially from the diamond.
This is in fact the bond that makes the bootstrap methodology fail at that maturity: The
price seems too high compared to what the Nelson Siegel model – and in fact all of the other
bonds around it – would imply. This fact suggests that either there is a trading opportunity
available, or that that data point is an aberration and should be corrected. Unfortunately,
such a correction is not easy if we use the bootstrap methodology.
Figure 2.6 The Term Structure of Interest Rates on December 31, 1993
5.5
5
Yield (%)
4.5
4
3.5
Bootstrap
Nelson Siegel
Extended Nelson Siegel
3
0.5
1
1.5
2
2.5
3
Time to Maturity
3.5
4
4.5
5
Source: Center for Research in Security Prices.
2.9.3.2 The Extended Nelson Siegel Model The Nelson Siegel model works well,
but it lacks the flexibility to match term structures that are highly nonlinear. The economist
Lars Svensson proposed an extension to the model, which is the one most widely adopted.
In particular, we assume:
1 − e− λ 1
T
r(0, T )
= θ0 + (θ1 + θ2 )
T
λ1
1 − e− λ 2
T
− θ2 e− λ 1 + θ3
T
T
λ2
− e− λ 2
T
(2.56)
where the parameters to estimate are 6: θi , i = 0, .., 3 and λ1 and λ2 . The procedure is
otherwise the same as in the case of the Nelson Siegel model. Figure 2.6 shows the results
of applying the extended Nelson Siegel model to the data in Table 2.6. The parameter
estimates are θ0 = 0.0687, θ1 = −0.0422, θ2 = −0.2399, θ3 = 0.2116, λ1 = 0.9652,
and λ2 = 0.8825. The outcome of the two procedures is almost the same. Indeed, the
extended Nelson Siegel model has been put forward to capture severe non-linearities in the
shape of the term structure of interest rates, a situation that did not occur in 1993.
70
BASICS OF FIXED INCOME SECURITIES
Figure 2.7
The Fit of the Nelson Siegel Model
125
data
Nelson Siegel
120
Bond Price
115
110
105
100
95
0
0.5
1
1.5
2
2.5
3
Time to Maturity
3.5
4
4.5
5
2.9.4 Curve Fitting with Splines
This is an extension of the curve fitting methodology described in Section 2.9.3, with a
different specification of the discount factor Z (t, T ) as a function of maturity T . In essence,
the idea is to assume that the discount function Z (t, T ) is given by a weighted average of
basis functions f (T ), where the weights are chosen to best match the bond prices.
Specifically, the discount function is given by
L
Z (t, T ) = 1 +
a f (T )
(2.57)
=1
What are the functions f (T )? Many alternatives have been proposed.
1. Simple polynomials:
f (T ) = T This is the simplest case, where the discount function is the Lth-order polynomial
L
Z (t, T ) = 1 +
a T =1
and the coefficients a have to be estimated to minimize the distance between observed
ni
c Tji ×
prices in the data Pci (0, Ti ) and the theoretical prices Pi (0, Ti ) = j =1
Z 0, Tji , where Tji is the jth-s cash-flow date of bond i. The problem with
polynomial functions is that they do not allow for a sufficient number of shapes,
APPENDIX: EXTRACTING THE DISCOUNT FACTORS Z (0, T ) FROM COUPON BONDS
71
without going into a very high order polynomial. In this case, however, the discount
function lacks the necessary stiffness to avoid being contaminated by small errors in
data.
2. Piecewise polynomial functions, or, spline: Intuitively, a polynomial spline can
be thought of as a number of separate polynomial functions, joined smoothly at a
number of so called “joints”, “breaks,” or “knot” points. Using this method, each
polynomial can be of low order and hence retain some stiffness, that is, a more
stable curve. Cubic splines are the most used functions (so, third order), as they
generate smooth forward curves. Of course, inside the family of splines there are
many specifications, such as (to give some names):
(a) Exponential cubic splines; and
(b) B-splines.
However, a number of other problems arise with these functions, the most important
being the decision of how many knot points to include and, in addition, where to
position them. We do not delve any more into this issue, as it is beyond the scope
of this chapter, but relevant readings are available in the references to this chapter at
the end of the book.
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CHAPTER 3
BASICS OF INTEREST RATE RISK
MANAGEMENT
Interest rates change substantially over time, and their variation poses large risks to financial institutions, portfolio managers, corporations, governments, and, in fact, households.
Anyone who directly or indirectly invests in fixed income securities or borrows money is
subject to interest rate risks. This chapter discusses the basics of interest rate risk management. In particular, we discuss first how to measure risk for fixed income instruments,
by introducing the notion of duration, value-at-risk and expected shortfall. Then, we also
cover the basic techniques to mitigate financial risk, such as immunization and asset liability
management.
3.1
THE VARIATION IN INTEREST RATES
Interest rates change substantially over time. Panel A of Figure 3.1 plots the time series
of yields from 1965 to 2005.1 The various lines, all very close to each other, are the
continuously compounded yields of zero coupon bonds for maturities from 1 month to 10
years. The most immediate fact that springs out from Panel A of this figure is that all yields
move up and down roughly together. For instance, they were all relatively low in the 1960s,
1 The
spot rate curves are calculated by fitting the extended Nelson Siegel model to coupon bond data from CRSP
(Monthly Treasuries) ¤2009 Center for Research in Security Prices (CRSP), The University of Chicago Booth
School of Business.
73
74
BASICS OF INTEREST RATE RISK MANAGEMENT
they were all relatively high in the late 1970s and early 1980s, and they were all relatively
low in the late 1990s.
Figure 3.1
Zero Coupon Bond Yields and the Level of Interest Rates: 1965 - 2005
A: Term Structure of Interest Rates
Interest Rate (%)
20
15
10
5
0
1965
1970
1975
1980
1985
1990
1995
2000
2005
1995
2000
2005
B: The Average Level of Zero Coupon Yields
Interest Rate (%)
20
15
10
5
0
1965
1970
1975
1980
1985
1990
Data Source: CRSP.
Panel B of Figure 3.1 plots the simple average of yields graphed in Panel A. We may
refer to this average generically as the level of interest rates.
Definition 3.1 The level of interest rates is the average yield across maturities.
As the level of interest rates changes over time, banks, bond portfolio managers and
corporations worry about the impact that the swings in interest rates have on the value of
their assets and liabilities. Two famous examples come to mind, namely, the savings and
loan crisis in the 1980s and the bankruptcy of Orange County, California, in 1994.
DURATION
3.1.1
75
The Savings and Loan Debacle
The savings and loan debacle in the 1980s is a standard example of what can go wrong when
interest rates shift. A savings and loan earns a large part of its revenues from the difference
between the long-term mortgages it provides to home owners and the short-term deposit
rate it offers to depositors. When interest rates increased at the end of the 1970s, savings
and loans were receiving their fixed coupons from mortgages contracted in the past, when
rates were low, but suddenly they had to pay interest on deposits at the new higher deposit
rates. Because depositors could choose where to put their money, banks were forced to
offer high deposit rates, otherwise depositors would withdraw their deposits and invest in
other securities, such as Treasuries. A withdrawal of funds is the worst nightmare for a
bank, as depositors’ money is not in the bank any longer: It has been loaned to others. The
spread between the rate earned on assets and the (higher) rate paid on liabilities quickly put
many savings and loans out of business.
3.1.2
The Bankruptcy of Orange County
In 1994 Orange County, California, lost $1.6 billion when the interest rate unexpectedly
increased from 3% to 5.7% over the course of the year.2 The substantial loss from the
total asset pool of $7.5 billion forced Orange County to declare bankruptcy. Through the
use of a mix of structure notes and leverage, Orange County’s portfolio stood to make
subtantial above market returns were the interest rate not to increase in the near future. But
interest rates did increase, and the fund went down. This famous case highlights yet another
example of the large losses that interest rate movements may bring about in portfolios that
are “too sensitive” to interest rates.
3.2
DURATION
The examples above calls for (a) a systematic methodology to assess the riskiness of a bond
portfolio to movements in interest rates; and (b) a methodology to effectively manage such
risk. We tackle the former problem in this section, and the latter in the next.
Definition 3.2 The duration of a security with price P is the (negative of the) percent
sensitivity of the price P to a small parallel shift in the level of interest rates. That is, let
r(t, T ) be the continuously compounded term structure of interest rates at time t. Consider
a uniform shift of size dr across rates that brings rates to r(t, T ), given by
r(t, T ) −→ r(t, T ) = r(t, T ) + dr
Let the price of the security move by dP as a consequence of the shift:
P −→ P = P + dP
2 See
the case study ERISK: Orange County, downloaded from the web site
http://www.erisk.com/Learning/CaseStudies/OrangeCounty.asp.
76
BASICS OF INTEREST RATE RISK MANAGEMENT
The duration of the asset is then defined as3
Duration = DP = −
1 dP
P dr
(3.1)
The shift dr is a small uniform change across maturities, such as, for instance, 1 basis
point: dr = .01%. The notion of duration then measures the impact that such a uniform
change on the yield curve has on the price of the security P . This can be seen by reorganizing
Equation 3.1 as follows:
Fact 3.1 Given a duration DP of a security with price P , a uniform change in the level of
interest rates brings about a change in the value of
Change in portfolio value = dP = −DP × P × dr
(3.2)
EXAMPLE 3.1
A $100 million bond portfolio has a duration equal 10, DP = 10. This implies that
one basis point increase in the level of interest rates dr = .01% generates a swing in
portfolio value of
Change in portfolio value = dP
= −10 × $100 million × .01/100
= −$100, 000
That is, the portfolio manager stands to lose $100,000 for every basis point increase
in the term structure.
How can we compute the duration of a security? Before we can answer this important
question, however, we need to recall the following two concepts from calculus. To simplify
our analysis, we will only consider continuously compounded interest rates, as in the
definition above. Below, we also review the more traditional notion of duration that uses
semi-annually compounded yield to maturity in its definition.
Definition 3.3 Let A and a be two constants and x be a variable. Let F (x) = A × eax
be a function of x. Then, the first derivative of F with respect to x, denoted by dF/dx, is
given by
Derivative of F (x) with respect to x =
dF
= A × a × eax = a × F (x)
dx
(3.3)
An example of the function F (x) is the zero coupon bond formula studied in Chapter 2
Pz (t, T ) = 100 × Z(t, T ) = 100 × e−r (T −t) .
3 The
duration definition in Equation 3.1 is often referred to as the “modified duration,” to differentiate it from the
Macaulay definition of duration, discussed below. In this book, we will rarely use the Macaulay duration, and
therefore we reserve the term duration for modified duration.
77
DURATION
In this case the constant A is the notional 100, the constant a equals the time to maturity
T − t, and the variable x equals the continuously compounded interest rate r. The notion
of the first derivative of Pz (t, T ) with respect to r then gives the sensitivity of the zero
coupon bond to the interest rate r.
Fact 3.2 Let Pz (r, t, T ) be the price of a zero coupon bond at time t with maturity T and
continuously compounded interest rate r. The first derivative of Pz (r, t, T ) with respect to
r is
d e−r (T −t)
d Pz
= 100 ×
dr
dr
= 100 × −(T − t) × e−r (T −t)
=
−(T − t) × Pz (r, t, T )
(3.4)
To emphasize that the zero coupon bond price depends on the current interest rate r,
in this section we denote it by Pz (r, t, T ), that is, we add r as one of the arguments in
Pz (t, T ). Visually, the first derivative represents the slope of the curve Pz (r, t, T ), plotted
against r, at the current interest rate level. More specifically, Figure 3.2 plots the price of
a 20-year zero coupon bond for various values of r, ranging from 0 to 15%. In the plot
T − t = 20, as the zero coupon bond has 20 years to maturity. Suppose today the interest
rate is r = 6%. The straight dotted line in the Figure is the tangent to the curve Pz (r, t, T )
at the point r = 6%. The slope of this tangent is the first derivative of Pz (r, t, T ) with
respect to r, dPz /dr.
3.2.1
Duration of a Zero Coupon Bond
We are now in the position of computing the duration of a zero coupon bond. The only
thing we have to remember is Definition 3.2, and the rule of the first derivative in Definition
3.3 when applied to a zero coupon bond (Equation 3.4). It is instructive to go through the
steps to compute the duration of a zero coupon bond, Dz ,T , where the notation “z” reminds
us that this calculation is done for a zero coupon bond.
dPz (r, t, T )
1
Dz ,T = −
(3.5)
Pz (r, t, T )
dr
1
× [−(T − t) × Pz (r, t, T )]
= −
Pz (r, t, T )
= T −t
(3.6)
The duration of a zero coupon bond is given by its time to maturity T − t. This makes it
very simple to compute, indeed.
EXAMPLE 3.2
A portfolio manager has $100 million invested in 5-year STRIPS. The duration of this
portfolio is then 5, implying that a one basis point increase in interest rates decreases
78
BASICS OF INTEREST RATE RISK MANAGEMENT
Figure 3.2
First Derivative of a Zero Coupon Bond with Respect to Interest Rate r
100
90
80
Zero Coupon Bond Pz(r,t;T)
70
60
50
40
dr
30
dP
20
Slope = d P / d r
10
0
0
5
10
15
Interest Rate r
the value of the portfolio approximately by
dP ≈ −DP × P × dr = −5 × $100 million × .01% = −$50, 000
3.2.2 Duration of a Portfolio
What is the duration of a portfolio of securities? Consider a portfolio made up of N1 units
of security 1, and N2 units of security 2. Let P1 and P2 be the prices of these two securities,
respectively. The value of the portfolio is then
W = N1 × P1 + N2 × P2
Let D1 and D2 be the duration of security 1 and security 2, respectively. By definition,
Di = −
1 d Pi
Pi d r
How can we determine the duration of the portfolio? We can apply the definition of duration
in Definition 3.2 and reorganize the expressions:
1 dW
W dr
d(N1 × P1 + N2 × P2 )
dr
dP1
dP2
+ N2 ×
N1 ×
dr
dr
Duration of portfolio = DW = −
1
W
1
= −
W
= −
(3.7)
79
DURATION
1
1 dP1
1 dP2
=
N1 × P 1 × −
+ N2 × P 2 × −
W
P1 d r
P2 d r
N2 × P 2
N1 × P 1
D1 +
D2
=
W
W
= w1 D1 + w2 D2
(3.8)
where
N1 × P 1
N2 × P 2
and w2 =
(3.9)
W
W
The expression in Equation 3.8 shows that the duration of a portfolio is a weighted average
of the durations of the assets, where the weights correspond to the percentage of the portfolio
invested in the given security.
w1 =
EXAMPLE 3.3
A bond portfolio manager has $100 million invested in 5-year STRIPS and $200
million invested in 10-year STRIPS. What is the impact of a one basis point parallel
shift of the term structure on the value of the portfolio?
We can answer this question by computing the duration of the portfolio: The
5-year and 10-year strips have duration of 5 and 10, respectively. The total portfolio
value is $300 million. Thus, the duration of the portfolio is
Duration of portfolio =
200
100
×5+
× 10 = 8.3
300
300
Therefore, a one basis point increase in interest rates generates a portfolio loss of
Loss in portfolio value = $300 million × 8.3 × 0.01% = $249, 000
Generalizing the formula in Equation 3.8 to n securities, we obtain:
Fact 3.3 The duration of portfolio of n securities is given by
n
DW =
wi Di
(3.10)
i= 1
where wi is the fraction of the portfolio invested in security i, and Di is the duration of
security i.
3.2.3
Duration of a Coupon Bond
We can apply the result in Fact 3.3 to compute the duration of a coupon bond. As discussed
in Section 2.4, a coupon bond with coupon rate c and n future coupon payments can be
considered a portfolio of zero coupon bonds, in which c/2 is invested in the first n − 1
zeros, and 1 + c/2 in the n−th zero:
n −1
Pc (0, Tn ) =
i= 1
c
c
× Pz (0, Ti ) + 1 +
Pz (0, Tn )
2
2
(3.11)
80
BASICS OF INTEREST RATE RISK MANAGEMENT
The duration of a coupon bond can then be computed by using Equation 3.10. Define the
weights
wi
=
c/2 × Pz (0, Ti )
for i = 1, .., n − 1
Pc (0, Tn )
wn
=
(1 + c/2) × Pz (0, Tn )
Pc (0, Tn )
Then, the duration of the coupon bond is
n
Dc
=
wi Dz ,T i
(3.12)
wi Ti
(3.13)
i= 1
n
=
i= 1
That is, the duration of a coupon bond is a weighted average of coupon payment times Ti .
EXAMPLE 3.4
Consider a 10-year, 6% coupon bond. Given a discount curve Z(0, T ), we can
compute its duration by following the calculations in Table 3.1. In this table, the
second and third columns present the payment times and the payment amounts. The
discount factor curve Z(0, T ) is in the fourth column. In the fifth column we compute
the discounted cash flows, the sum of which gives the price Pc (0, T ) = $107.795 at
the bottom of the table. The weights in column 6 equal the discount cash flows in
column 5 divided by the price. Finally, the last column reports the weighted payment
times: ω i × Ti . The duration is given by the sum of these weighted payment times,
reported at the bottom of the table: Dc = 7.762. Different from zero coupon bonds,
the duration of the coupon bond is shorter than its maturity.
3.2.4 Duration and Average Time of Cash Flow Payments
While we have derived the formula for duration in Equation 3.13 from the definition
of duration as the percentage sensitivity of a security to changes in interest rates (see
Definition 3.2), some confusion sometimes arises about the notion of duration because
sometimes people define duration as the average time of payments, as in Equation 3.13
(see also Section 3.2.6). These two interpretations are equivalent for fixed rate bonds,
that is, bonds that pay fixed coupons: A zero coupon bond with maturity of 5 years has
duration equal to 5. This is both the average time of payments (there is only one), and
also the percentage loss in value from an increase in interest rates. A similar situation
exists for coupon bonds. However, for many securities that do not have fixed payments,
the equivalence is broken. The following provides a simple example:
DURATION
81
Table 3.1 Duration of Coupon Bond, Coupon = 6%
Period
i
Payment
Time
Ti
Cash
Flow
CF
Discount
Z(0, Ti )
Discounted
Cash Flow
CF ×Z(0, T0 )
Weight
wi
Weight ×Ti
w i × Ti
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
103
0.976
0.952
0.929
0.906
0.884
0.862
0.841
0.821
0.801
0.781
0.762
0.744
0.725
0.708
0.690
0.674
0.657
0.641
0.626
0.610
2.927
2.855
2.786
2.718
2.652
2.587
2.524
2.462
2.402
2.344
2.286
2.231
2.176
2.123
2.071
2.021
1.972
1.923
1.877
62.858
0.027
0.026
0.026
0.025
0.025
0.024
0.023
0.023
0.022
0.022
0.021
0.021
0.020
0.020
0.019
0.019
0.018
0.018
0.017
0.583
0.014
0.026
0.039
0.050
0.061
0.072
0.082
0.091
0.100
0.109
0.117
0.124
0.131
0.138
0.144
0.150
0.155
0.161
0.165
5.831
Price
107.795
Duration
7.762
82
BASICS OF INTEREST RATE RISK MANAGEMENT
EXAMPLE 3.5
Recall that in Section 2.5 of Chapter 2 we considered the price of a floating rate bond.
In particular, Fact 2.12 shows that if Ti denotes the last reset date, Ti+1 denotes the
next reset date, t is the current time, with Ti < t < Ti+1 , then the price of a floating
rate bond with maturity T and semi-annual payments is given by (see Equation 2.40)
PF R (t, T ) = Z(t, Ti+ 1 ) × 100 × [1 + r2 (Ti )/2]
(3.14)
where r2 (Ti ) is the reference rate that is determined at the last reset date. The duration
of the floating rate bond is then
Duration of
floating rate
bond at t
d PF R
1
(3.15)
PF R (t, T ) d r
1
d Z(t, Ti+1 )
r2 (Ti )
= −
× 100 × 1 +
PF R (t, T )
dr
2
1
r2 (Ti )
[−(Ti+ 1 − t)] × Z(t, Ti+1 ) × 100 × 1 +
= −
PF R (t, T )
2
(3.16)
= Ti − t
= DF R = −
where the last equality stems from using again Equation 3.14. Equation 3.16 shows
that the duration of a floating rate bond is simply equal to the time left to the next
coupon payment date Ti+ 1 − t. In particular, if today is coupon date (but the coupon
has not been paid yet), the duration is zero.
This example shows that even if the average time of future cash flows can be relatively long
– a floating coupon bond with 10 years to maturity, for instance, has an average time of
future payments of several years – the duration could be very small. Conversely, we will see
securities for which the duration is actually longer than their maturity, or securities for which
the duration is negative. Given that in modern times the notion of duration is mainly used
for risk management purposes, and in particular to compute the sensitivity of a security to
parallel shifts in the term structure, we must be careful in interpreting duration as an average
time of future payments, as this interpretation only holds for securities with fixed cash flows.
3.2.5 Properties of Duration
It is important to realize that the duration of a coupon bond depends crucially on the level
of the coupon rate. As the coupon rate increases, the duration is lower. The first three
columns of Table 3.2 show this effect for the case in Example 3.4. What is the intuition?
There are two ways to see this result intuitively:
1. Lower Average Time of Cash Flow Payments: The higher the coupon, the larger are
the intermediate coupons relative to the last one (in 10 years). Thus, the average time
of coupon payments gets closer to today.
2. Lower Sensitivity to Interest Rates: The higher the coupon rate, the larger are cash
flows in the near future compared to the long-term future. Cash flows that arrive
sooner rather than later are less sensitive to changes in interest rates (for instance, a
DURATION
83
Table 3.2 Duration versus Coupon Rate and Interest Rate
Coupon
c
Price
Pc
Duration
D
Interest Rate
r2
Price
Pc
Duration
D
0
2%
4%
6%
8%
10%
12%
61.03
76.62
92.21
107.79
123.38
138.97
154.56
10.00
8.95
8.26
7.76
7.39
7.11
6.88
1%
3%
5%
7%
9%
11%
13%
147.47
125.75
107.79
92.89
80.49
70.12
61.44
8.13
7.95
7.76
7.56
7.35
7.12
6.90
cash flow arriving tomorrow has no sensitivity to interest rates). Thus, an increase in
coupon rate implies an overall lower sensitivity to changes in discount rates.
For the same reason, the duration of a coupon bond decreases with the general level of
interest rates, as we see for the last three columns of Table 3.2. In this case, the coupon rate
is kept at 6%, but the semi-annual interest rate r2 – constant across maturities – increases
from 1% to 13%. Note that the duration drops from 8.13 to 6.90. Once again, a higher
interest rate (across maturities) implies that short-term cash flows have a relatively higher
weight in the value of the bond, and thus a lower sensitivity to changes in interest rates.
3.2.6
Traditional Definitions of Duration
We defined the duration as
1 dP
(3.17)
P dr
where r is the continuously compounded interest rate. This definition of duration is simple
to apply in order to compute the duration of interest rate securities, from zero coupon
bonds to portfolios of securities. For instance, Equation 3.17 shows that the duration of a
fixed coupon bond equals the average time of payment times, which is a relatively simple
formula to determine the sensitivity of a coupon bond to parallel shifts in the yield curve.
Traditionally, however, the duration is not defined against the continuously compounded
interest rate but rather against the semi-annually compounded yield to maturity. In this
case, the definition of the modified duration as the (negative of the) sensitivity of prices to
changes in interest rates (Equation 3.17) does not correspond exactly to the simple formulas
derived earlier, and a small adjustment is needed. In particular, consider a security with
yield to maturity y (see Section 2.4.3 in Chapter 2). Recall that by definition of yield to
maturity, the price of the coupon bond on a coupon date can be written as
D=−
n
Pc (0, T ) =
j=1
c/2 × 100
100
+
2×T n
y 2×T j
1+ 2
1 + y2
(3.18)
A little algebra shows that the modified duration (M D) of this coupon bond, when defined
against the yield to maturity y, is given by
MD = −
1
1 dP
=
P dy
1 + y2
n
wj × Tj
j =1
(3.19)
84
BASICS OF INTEREST RATE RISK MANAGEMENT
where
wj =
1
Pc (0, T )
c/2 × 100
2×T j
1 + y2
, wn =
1
Pc (0, T )
100 × (c/2 + 1)
2×T n
1 + y2
In other words, when we use the semi-annual compounded yield to maturity y to define the
modified duration, then the modified duration of a fixed rate bond can be computed as the
weighted average of cash flow maturities times an adjustment given by (1/(1 + y2 ). The
weighted average of cash flow maturities in Equation 3.19 is called the Macaulay duration
n
DM c =
wj × Tj
(3.20)
j=1
We will rarely use the variation in the semi-annually compounded yield to maturity for
risk management purposes, and rather use the variation in the continuously compounded
spot curve. Not only does this choice allow for simpler formulas, as we showed in the
previous sections, but it also implies that the durations of different assets are defined against
the variation of the same interest rates, namely, the spot rates. Instead, a definition in terms
of yield to maturity hinges on the notion of yield to maturity itself, which, as discussed
in Chapter 2 (Section 2.4.3) has some issues, such as the fact that it is bond specific, it
depends on the coupon rate, and so on. In addition, for several fixed income securities the
notion of yield to maturity is not well defined, because they may have floating rate coupons
or embedded options. The notion of a parallel shift in the spot curve is always well defined
for any interest rate security, and whenever an analytical formula is not available, we can
always rely on computers to obtain an approximate quantity, called effective duration. We
apply such a strategy for instance in Chapter 8 in the context of mortgage backed securities.
3.2.7 The Duration of Zero Investment Portfolios: Dollar Duration
The definition of duration in Equation 3.1 implicitly implies that the security, or the
portfolio, has nonzero value. However, in many interesting cases involving no arbitrage
strategies, the security or the portfolio may have a value of exactly zero. In this case, we
resort to the dollar duration:
Definition 3.4 The dollar duration D$ of a security P is defined by
Dollar duration = DP$ = −
dP
dr
(3.21)
That is, the dollar duration corresponds to the (negative of the) sensitivity of the price P
to changes in the level of interest rate r. Since dP is the change in the price of the security,
in dollars, the name dollar duration follows. Combining Equations 3.1 and 3.21 we obtain:
Fact 3.4 For a nonzero valued security or portfolio with price P , the relation between
duration and dollar duration is
DP$ = P × DP
(3.22)
85
DURATION
In this case, the relation between the dollar duration of the portfolio and the dollar
duration of its individual components is given by the sum of the dollar durations:
$
is given by
Fact 3.5 The dollar duration of portfolio of n securities, denoted by DW
n
$
DW
=
Ni Di$
(3.23)
i= 1
where Ni is the number of units of security i in the portfolio, and Di$ is the dollar duration
of security i.
EXAMPLE 3.6
Dollar Duration of a Long-Short Strategy
Let the term structure of interest rates be flat at 4% (semi-annually compounded).
Consider an arbitrageur who is contemplating going long a 4% coupon bond by
borrowing at the current floating rate. To keep the analysis simple, suppose the
arbitrageur can enter into term repos with maturity of six months and that the haircut
is zero. Because the term structure of interest rate is flat, a 4% coupon bond would
be valued at par ($100), which is the amount the arbitrageur needs to borrow. The
portfolio has value of zero at time t = 0 when the trade is set up. However, the trade
is risky, because if interest rates move up, then the arbitrageur will suffer a decrease
in value in the long position that is not paralleled by an equivalent decrease in value
in the short (borrowing) position.
More specifically, borrowing at the 6-month term repo is essentially equivalent to
shorting a 6-month floating rate bond. Thus, the long-short portfolio can be written
as
W = Pc (0, T ) − PF R (0, T ) = 0
Let the duration of the fixed rate bond be 8.34. The duration of the floating rate bond
is 6 months, as discussed in Example 3.5. Thus, using Equation 3.22, we find
Dollar duration of Pc (0, T )
Dollar duration of PF R (0, T )
=
$100 × Duration of fixed rate bond
=
$100 × 8.34 = $834
=
$100 × Duration of floating rate bond
=
$100 × 0.5 = $50
Thus, the dollar duration of the long-short portfolio is
Dollar duration of long-short portfolio = $834 − $50 = $784
(3.24)
Again using Equation 3.21, we have that one basis point increase in interest rate dr
generates change in the long-short portfolio:
$
Change in portfolio value = d W = −DW
× d r = −$784 × .01/100 = −.0784 (3.25)
That is, the long-short portfolio with trade size of $1 million, for instance, stands to
lose $78,400 for every basis point increase in the level of interest rates.
86
BASICS OF INTEREST RATE RISK MANAGEMENT
The dollar losses due to a basis point increase in the level of interest rates, as computed in
Equation 3.25 is a common measure of interest rate risk. Traders refer to it as the “price
value of a basis point,” or PVBP, or PV01:
Definition 3.5 The price value of a basis point PV01 of a security with price P is defined
as
(3.26)
Price value of a basis point = P V 01(orP V BP ) = −DP$ × d r
3.2.8 Duration and Value-at-Risk
Value-at-Risk (VaR) is a risk measure that attempts to quantify the amount of risk in a
portfolio. In brief, VaR answers the following question: With 95% probability, what is the
maximum portfolio loss that we can expect within a given horizon, such as a day, a week
or a month? Methodologies for the computation of VaR are many and range from highly
complex to fairly simple. In this section we discuss two methodologies that are based on
the concept of duration: The historical distribution approach and the normal distribution
approach.
Definition 3.6
Let α be a percentile (e.g. 5%) and T a given horizon. The (100−α)% T Value-at-Risk
of a portfolio P is the maximum loss the portfolio can suffer over the horizon T with α%
probability. In formulas, let LT = −(PT − P0 ) denote the loss of a portfolio over the
horizon T (a profit if negative). The VaR is that number such that:
P rob (LT > V aR) = α%
(3.27)
For instance, a $100 million bond portfolio may have a 95%, 1-month VaR of $3 million.
This VaR measure implies that there is only 5% probability that the portfolio losses will be
higher than $3 million over the next month.
The VaR measure is based on the volatility of the underlying assets in the portfolio. For
bond portfolios, the volatility is determined by movements in the interest rates. In fact,
through duration, we can estimate the sensitivity of a portfolio to fluctuations in the interest
rate. Recall Equation 3.2:
(3.28)
dP = −DP × P × dr
Given the value of the portfolio P and its duration DP , we can transform the probability
distribution of interest rate changes dr into the probability distribution of portfolio changes
dP , and from the latter, we can compute the portfolio potential losses. The 95% VaR
corresponds to the 5% worst case of distribution of dP . A simple example is given by the
case in which dr has a normal distribution:
Fact 3.6 Let dr have a normal distribution with mean μ and standard deviation σ. Then
Equation 3.28 implies that dP has a normal distribution with mean and standard deviation
given by:
(3.29)
μP = −DP × P × μ and σ P = DP × P × σ.
87
DURATION
That is:
dr ∼ N (μ, σ 2 )
=⇒
dP ∼ N (μP , σ 2P )
(3.30)
The 95% VaR is then given by
95% VaR = −(μP − 1.645 × σ P )
(3.31)
where −1.645 corresponds to the 5-th percentile of the standard normal distribution, that
is, if x ∼ N (0, 1) then P rob(x < −1.645) = 5%. The 99% VaR is computed as in
Equation 3.31 except that the number “1.645” is substituted by “2.326.”
This result of course relies on Equation 3.28, which is only an approximation. If dr is not
normal, Equation 3.31 does not hold. The next example illustrates one popular approach
to dealing with this latter case.
EXAMPLE 3.7
A portfolio manager has $100 million invested in a bond portfolio with duration
DP = 5. What is the 95% one-month Value-at-Risk of the portfolio?
1. Historical Distribution Approach. We can use the past changes in the level of
interest rates dr as a basis to evaluate the potential changes in a portfolio value dP .
Panel B of Figure 3.1 shows the historical observations of the level of interest rates
up to 2005. Panel A of Figure 3.3 shows the monthly changes in the level of interest
rates, while Panel B makes a histogram of these variations. As we can see large
increases and decreases are not very likely, but they do occur occasionally. We can
now multiply each of these changes dr observed in the plot by −DP ×P = −5×100
million to obtain the variation in dP . Panel C of Figure 3.3 plots the histogram of
the changes in the portfolio i.e., the portfolio profits and losses (P&L).4 Given this
distribution, we can compute the maximum loss that can occur with 95% probability.
We can start from the left-hand side of the distribution, and move rightward until
we count 5% of the observations. That number is the 95% monthly VaR computed
using the historical distribution approach. In this case, we find it equal to $3 million.
That is, there is only 5% probability that the portfolio losses will be higher than $3
million.
2. Normal Distribution Approach. From Fact 3.6, a normal distribution assumptin on
dr translates into a normal distribution on dP . Using the data plotted in Panel A of
Figure 3.3, we find that the monthly change in interest rate has mean μ = 6.5197 ×
10 − 006 and stadard deviation σ = .4153%. Therefore, μP = −5 × 100 × μ =
−.0033 and σ P = 5×100×σ = 2.0767. The standard normal distribution is plotted
along with the (renormalized) histogram in Panel C of Figure 3.3. The 95% VaR is
then equal to 95% VaR = −(μP − 1.645 × σ P ) = $3.4194 million.
4 We
renormalized the histogram to make it comparable with the normal distribution case, discussed in the next
point.
88
BASICS OF INTEREST RATE RISK MANAGEMENT
Figure 3.3 Changes in the Level of Interest Rates: 1965 - 2005
Interest Rate (%)
A: Monthly Changes in the Level of Interest Rates
4
2
0
−2
−4
1965
1970
1975
1980
1985
1990
1995
2000
2005
2
3
4
10
15
20
B: Histogram of Monthly Changes in the Level of Interest Rates
Occurrences
40
30
20
10
Probability Density
0
−4
−3
−2
−15
−10
−1
0
1
Interest Rate (%)
C: Probability Distribution of Portfolio P/L
0.4
0.3
0.2
0.1
0
−20
−5
0
Millions of Dollar
5
Data Source: CRSP.
3.2.8.1 Warnings It is worth emphasizing immediately a few problems with the
Value-at-Risk measure of risk, as well as some potential pitfalls:
1. VaR is a statistical measure of risk, and as with any other statistical measure, it
depends on distributional assumptions and the sample used for the calculation. The
difference can be large. For instance, in Example 3.7 the VaR varies depending
on whether we use the normal distribution approach or the historical distribution
approach.
2. The duration approximation in Equation 3.28 is appropriate for small parallel changes
in the level of interest rates. However, by definition, VaR is concerned with large
changes. Therefore, the duration approximation method is internally inconsistent.
DURATION
89
The problem turns out to be especially severe for portfolios that include derivative
securities, either implicitly or explicitly. We will return to this issue in later chapters.
3. The VaR measures the maximum loss with 95% probability. However, it does not
say anything about how large the losses could be if they do occur. The tails of the
probability distribution matter for risk. For instance, in Example 3.7 the 99% VaR
using the historical distribution approach is $5.52 million, while this figure is only
$4.83 million using the normal distribution assumption. The tails of the normal
distribution are thin, in the sense that they give an extremely low probability to large
events, which instead in reality occur with some frequency.
4. The VaR formula used in Equation 3.31 includes the expected change in the portfolio
μP = −DP ×P ×E[dr]. The computation on the expected change E[dr] is typically
very imprecise, and standard errors are large. Such errors can generate a large error
in the VaR computation. For this reason, it is often more accurate to consider only
the unexpected VaR, that is, consider only the 95% loss compared to the expected
P&L μP . Practically, we simply need to set μP = 0 in Equation 3.31.
3.2.9
Duration and Expected Shortfall
Some of the problems with VaR can be solved by using a different measure of risk, called
the expected shortfall. This measure of risk answers the following question: How large can
we expect the loss of a portfolio to be when it is higher than VaR? As mentioned in point 3 in
the above Subsection 3.2.8.1, the VaR measure does not say anything about the tails of the
statistical distribution. This is an especially important problem when the underlying risk
factor has a fat-tailed distribution, as shown in Figure 3.3, or when the portfolio contains
highly nonlinear derivative securities, as we will see in later chapters.
Definition 3.7 The expected shortfall is the expected loss on a portfolio P over the horizon
T conditional on the loss being larger than the (100 − α)%, T VaR:
Expected shortfall = E [LT |LT > V aR]
(3.32)
For instance, a $100 million portfolio may have a 95%, 1-month expected shortfall of
$4.28 million, meaning that when a bad event hits (losses higher than VaR), the portfolio
stands to lose $4.28 million in average.
The calculation of expected shortfall is only slightly more involved than the one of VaR.
For instance, for normally distributed variables, we have the following result:
Fact 3.7 Consider Fact 3.6. Under these assumptions:
f (−1.645)
95% Expected shortfall = − μP − σ P ×
N (−1.645)
= − (μP − σ P × 2.0628)
(3.33)
(3.34)
90
BASICS OF INTEREST RATE RISK MANAGEMENT
where f (x) denotes the standard normal density and N (x) is the standard normal cumulative density,5 The 99% expected shortfall is obtained as in Equation 3.34 except with the
number “2.6649” in place of “2.0628.”
A quick comparison of Equations 3.34 and 3.31 shows that for the normal distribution
case, the expected shortfall contains the same information as the Value-at-Risk, as the only
difference is the coefficient that multiplies σ P . But this is in fact exactly the reason for a
new measure of risk: The expected shortfall is very useful precisely for those situations in
which the portfolio losses are not expected to be normally distributed.
EXAMPLE 3.8
Consider again Example 3.7. The 95%, 1-month expected shortfall is easily computed
in the case of a normal distribution, as we must simply change the coefficient “1.645”
that multiplies σ P with the coefficient “2.0628” (and similarly for the 99% expected
shortfall). Given μP = −.0033 and σ P = 2.0767, we obtain
95% ES = $4.2871 mil; 99% ES = $5.5374 mil
(3.35)
The numbers are quite different for the case in which the historical distribution
approach is used. How do we compute the expected shortfall in this case? The
methodology is just a slight modification of the VaR computation. In the VaR case,
we first rank all of the portfolio P&L realizations under the various interest rate
scenarios from the worst to the best, and then pick the 5% worst case. For the
expected shortfall, we take the average of all of the realizations below the 5% worst
case. A similar precedure is used for the 1% expected shortfall calculation. In this
case, we obtain:
(Normal disttribution approach):
95% ES = $5.0709 mil; 99% ES = $9.3344 mil
(3.36)
We note in particular that the 99% expected shortfall is substantially larger under
the historical distribution approach than under the normal distribution approach. This
finding is a reflection of the fat-tailed distribution that characterizes the interest rate
changes, and thus of the P&L dP , as shown in the bottom panel in Figure 3.3. In
particular, extreme portfolio realizations occur more frequently than according to the
normal distribution. It is worth pointing out that in contrast the VaR measure does
not capture well the risk embedded in the tails of the distribution. For instance, in
Example 3.7 the 99% VaR is $5.52 million, which is higher than the figure obtained
under the normal distribution approach ($4.83 million), but not much higher. The
expected shortfall is much better able to capture the risk from tail events.
(Historical distribution approach):
3.3 INTEREST RATE RISK MANAGEMENT
Interest rate risk management is a key activity for banks, bond portfolio managers, corporations, governments, and, in fact, households. To understand the risks in interest rate
fluctuations, consider the following example.
5 That
√
2
is, f (x) = 1/ 2π × e−x / 2 and N (x) =
x
−∞
f (y)dy.
91
INTEREST RATE RISK MANAGEMENT
EXAMPLE 3.9
Ms. Caselli retired at the age of 60, with $1,000,000 in her retirement account.
She now has to decide where to invest this amount of money for the next, say, 30
years. Treasury bonds are the only type of security she would consider, given her
age. Should she invest in long-term bonds or short-term bonds? Consider the two
extremes:
1. Invest all of $1,000,000 in 6-months T-bills.
2. Invest all of $1,000,000 in 30-year T-bonds.
What is the difference between these two strategies? If Ms. Caselli is going to
consume only the interest on her investment, strategy 1 is more risky than strategy
2. Indeed, under strategy 1, fluctuations in interest rates imply fluctuations in the
amount of money available for consumption. For instance, an interest rate change
from 4% to 1% decreases Ms. Caselli’s annual interest income from $40,000 to
$10,000, a rather dramatic change. Instead, assuming that the 30-year bond sells at
par and that the coupon rate is 4%, strategy 2 provides a certain $40,000 per year for
all 30 years.
Most likely, Ms. Caselli is interested in using up some of her capital for consumption purposes. Indeed, cashing nothing but interest income may not produce
enough funds on which to survive. If cash flow comes from the amount of capital
available, the sensitivity of capital itself to interest rates becomes a big issue. For
instance, look again at strategy 2. Assume that the zero coupon yield curve is flat at
4% (semi-annually compounded), so that a 30-year T-bond with 4% coupon trades
at par. Such coupon bond has a duration of 17.72. Consider now an interest rate
increase of 3% from 4% to 7% (as happened, for instance, in 1994). The capital
losses on the investment would be approximately
Capital losses ≈ 17.72 × $1 million × .03 = $531, 000
That is, a 3 percent increase in the interest rate may more than halve the savings of
Ms. Caselli. If Ms. Caselli is not planning to consume out of her capital, this capital
loss is of no consequence: She still possesses the same bond as before, which will
keep paying the same $40,000 per year. But if she is planning to use up some of the
capital for consumption, this strategy is clearly risky.
This example illustrates how the type of interest rate risk management that an institution or
a person may want to engage in depends on the goals of the institution or individual.
3.3.1
Cash Flow Matching and Immunization
Ms. Caselli, in Example 3.9, can purchase an annuity from a financial institution. For
instance, the financial institution may agree to pay $28,767 every six months for 30 years
in exchange for the $1,000,000 deposit. Where is this number coming from? Assuming a
flat term structure at a semi-annually compounded interest rate of 4%, the present value of
this stream of cash flows is about $1,000,000:
60
$1, 000, 000 = $28, 767 ×
i=1
1
1+
4%
2
i
(3.37)
92
BASICS OF INTEREST RATE RISK MANAGEMENT
where 60 is the number of payments.
How can the financial institution now hedge this commitment to pay exactly $28,767
twice a year for 30 years? What risks does it take?
1. Cash Flow Matching. The financial institution can purchase a set of securities that
pays exactly $28,767 every six months. For instance, it can purchase 60 zero coupon
bonds, each with a $28,767 face value, and with maturities of 6 months, 1 year, 1.5
years, and so on up to 30 years. The value of these securities is, by construction,
equal to $1,000,000, as the present value in Equation 3.37 applies to this case.
One drawback of this strategy, though, is that the financial institution should find
exactly the type of securities that are required for the cash flow matching, such as the
sequence of zero coupon bonds with $28,500 face value at 6-month intervals. Such
an endeavor may be problematic, and costly, as many securities are nonliquid.
2. Immunization. The financial institution can engage in a dynamic immunization
strategy. Such a strategy involves the choice of a portfolio of securities with the same
present value and duration of the cash flow commitments to pay. Immunization is
preferred over cash flow matching as it allows the institution to choose bonds that
have favorable properties in terms of liquidity and transaction costs. If executed
properly, this method generates the desired stream of cash flows.
While the cash flow matching is relatively straightforward, it is instructive to work
through an example illustrating the immunization strategy. We continue with Example 3.9.
EXAMPLE 3.10
The financial institution that took up the commitment to pay $28,767 every six months
can ensure the ability to pay by engaging in the following dynamic strategy. Let xt %
denote the fraction of the total capital – $1,000,000 at initiation – invested in the 4%,
30-year bond, as described in Example 3.9. Assume that the remaining (1 − xt )%
is kept as cash in a deposit account, thereby yielding the overnight deposit rate. The
duration of the annuity promised to Ms. Caselli is about 12.35. The 30-year coupon
bond has a duration of 17.72, while the overnight deposit has zero duration, as the
deposit rate resets daily. Because the immunization strategy calls for equating the
duration of the portfolio with the one of the annuity, it then requires that at time 0:
x0 % × 17.72 + (1 − x0 %) × 0 = 12.34 =⇒ x0 = 71%
Assume that the financial institution rebalances every six months. Then, at any time
t = .5, 1, 1.5, ..., 30 the financial institution:
• collects the
4%
2
coupon from the 30-year bond;
• collects the interest cumulated over the six months on the cash deposit;
• pays the annuity cash flow of $28,767 to Ms. Caselli; and
• reinvests the remaining balance in long-term bonds and overnight deposit according
to the rule:
Duration of annuity
Percentage investment in long-term bond = xt =
Duration of long-term bond.
(3.38)
93
INTEREST RATE RISK MANAGEMENT
Table 3.3 illustrates the strategy. Column (1) reports the time at which coupon
payments are made, and rebalancing takes place. For convenience, assume that
annuity payments and the long-term bond coupon payments occur on the same date.
Column (2) reports a possible path of interest rates, from 4% to 11% and down again
to 8% in the course of 30 years. These interest rates have been simulated. Column
(3) computes the balance of the financial institution. It starts out with $1 million, and
then the balance declines as the financial institution makes coupon payments to Ms.
Caselli. We are more explicit about the information in this column below. Column
(4) reports the present value of the annuity, assuming that the term structure is flat
and equal to the interest rate in Column (2). Column (5) indicates the duration of
the annuity. Note that both the present value and the duration of the annuity tend
to decline over time. Columns (6) and (7) report the present value and duration,
respectively, of the 4%, 30-year T-bond that is used in the immunization strategy.
Column (8) reports the fraction of capital xt invested in the 30-year bond, obtained
from using Equation 3.38. Column (9) shows the total cash obtained at the end of
each six-month period t, from the investment in overnight deposits at the beginning
of the period. That is,
Interest payment [column (9)] = Wt × (1 − xt ) × rt /2
(3.39)
Similarly, Column (10) represents the total coupon received from the 30-year bond
investment
Coupon payment [column (10)] =
Wt × xt
× 4%/2
Price T-bond in Column (6)
(3.40)
Finally, returning to Column (3), the total amount of capital at the institution is
updated by taking into account inflows and outflows. That is
Wt+ 1
= Wt × (1 − xt ) + Wt × xt × Capital gain on T-bond
(3.41)
+ Interest in (9) at t + coupon in (10)
(3.42)
− Annuity coupon ($28,767)
(3.43)
Notice from the last row in Table 3.3 that the strategy still leaves $69,375 at
maturity. If the interest rate was constant for the overall period and equal to 4%, then
the final amount of wealth WT would be exactly zero. There is a reason why the
final wealth came up positive – due to the convexity of bond prices with respect to
interest rates – that we discuss in Chapter 4.
Was this luck? That is, if we consider a different path of interest rates, would
we still get a positive number? Or are there instances in which the final wealth was
negative? To convince ourselves that this dynamic immunization strategy works, we
can repeat the above exercise many times, for many interest rate scenarios, and plot
the histogram of the final value WT . This is done in Figure 3.4. As it can be seen,
the strategy works well, as the final wealth is always positive.
3.3.2
Immunization versus Simpler Investment Strategies
How does the immunization strategy compare to other simpler strategies, such as
investing fixed proportions in the long-term T-bond and cash? The panels in Figure
94
BASICS OF INTEREST RATE RISK MANAGEMENT
Table 3.3
(1)
(2)
(3)
t
r2
Wt
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
13.5
14.0
14.5
15.0
15.5
16.0
16.5
17.0
17.5
18.0
18.5
19.0
19.5
20.0
20.5
21.0
21.5
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
27.0
27.5
28.0
28.5
29.0
29.5
30.0
4.00%
4.53%
5.46%
5.80%
5.07%
5.70%
5.97%
5.51%
5.75%
5.62%
5.41%
4.44%
3.84%
4.37%
4.85%
5.22%
5.80%
6.21%
7.10%
7.90%
8.77%
8.00%
8.34%
7.91%
7.96%
8.59%
9.55%
9.27%
10.09%
10.49%
10.19%
10.10%
10.84%
11.34%
12.08%
11.95%
11.58%
11.45%
11.63%
11.94%
11.82%
11.69%
11.35%
11.48%
11.35%
10.98%
10.15%
10.30%
10.72%
10.32%
9.51%
9.97%
10.15%
9.54%
10.38%
10.79%
10.12%
9.64%
8.99%
8.30%
1,000,000.00
931,694.92
834,369.76
798,630.50
858,995.30
798,169.63
770,555.63
802,061.06
775,789.15
780,072.25
789,534.15
863,655.83
909,537.36
851,902.96
803,888.62
767,558.29
720,606.80
688,142.41
633,131.27
589,160.97
547,447.66
574,852.24
555,509.40
567,397.43
559,108.10
529,197.45
491,704.85
495,447.31
465,514.05
449,693.51
452,268.71
448,755.10
425,079.56
408,561.98
388,211.13
385,320.32
386,506.22
382,431.96
372,563.61
360,381.56
354,983.36
349,068.99
345,393.43
334,579.86
326,472.31
320,443.04
318,131.85
303,990.10
287,314.17
276,993.67
268,191.16
249,375.96
232,116.30
217,795.73
196,476.08
176,812.76
158,948.97
138,908.31
117,422.88
94,234.40
69,375.15
Example of Immunization Strategy
(4)
PV
Annuity
(5)
D
Annuity
(6)
PV
T-bond
(7)
D
T-bond
(8)
xt
(9)
Interest
Payment
(10)
Coupon
T-bond
1,000,000.00
931,527.35
833,021.75
797,418.94
855,333.53
794,486.32
767,029.58
797,212.37
771,099.47
774,970.46
783,823.66
853,748.21
896,984.46
839,884.97
792,339.74
756,355.88
709,848.70
677,682.96
623,058.85
579,458.13
538,080.12
563,207.45
543,902.64
554,342.75
545,629.81
516,057.51
479,048.80
481,641.25
452,081.69
436,142.65
437,477.20
433,060.02
409,594.85
392,937.71
372,656.40
368,625.65
368,186.27
362,838.04
352,126.89
339,248.28
332,387.26
324,921.21
319,247.45
307,161.34
297,276.29
288,950.85
283,432.61
267,790.08
249,904.43
236,955.12
224,811.75
204,603.79
185,357.01
167,873.19
145,132.50
123,208.88
101,873.37
78,608.20
53,876.71
27,622.23
12.35
11.84
11.12
10.79
11.08
10.59
10.32
10.43
10.17
10.10
10.05
10.34
10.45
10.04
9.68
9.37
9.00
8.71
8.27
7.90
7.52
7.61
7.39
7.36
7.21
6.93
6.59
6.52
6.23
6.04
5.95
5.83
5.59
5.39
5.16
5.04
4.94
4.81
4.64
4.46
4.32
4.16
4.02
3.84
3.67
3.51
3.35
3.15
2.94
2.75
2.56
2.34
2.12
1.91
1.68
1.45
1.22
0.98
0.74
0.50
1.00
0.91
0.79
0.75
0.84
0.77
0.74
0.79
0.76
0.78
0.81
0.93
1.03
0.95
0.88
0.84
0.78
0.74
0.66
0.61
0.55
0.61
0.59
0.62
0.62
0.59
0.54
0.56
0.52
0.51
0.53
0.54
0.51
0.50
0.48
0.49
0.52
0.53
0.53
0.53
0.55
0.57
0.59
0.60
0.62
0.65
0.70
0.71
0.71
0.74
0.78
0.79
0.80
0.84
0.84
0.85
0.89
0.92
0.95
0.98
1.00
17.73
17.00
15.88
15.42
16.03
15.31
14.94
15.24
14.90
14.89
14.92
15.51
15.76
15.19
14.69
14.28
13.75
13.35
12.70
12.14
11.57
11.84
11.56
11.62
11.45
11.05
10.55
10.52
10.11
9.86
9.80
9.67
9.33
9.06
8.74
8.60
8.49
8.32
8.08
7.82
7.60
7.37
7.13
6.84
6.56
6.26
5.97
5.62
5.25
4.89
4.51
4.11
3.70
3.28
2.84
2.39
1.94
1.47
0.99
0.50
0.70
0.70
0.70
0.70
0.69
0.69
0.69
0.68
0.68
0.68
0.67
0.67
0.66
0.66
0.66
0.66
0.65
0.65
0.65
0.65
0.65
0.64
0.64
0.63
0.63
0.63
0.62
0.62
0.62
0.61
0.61
0.60
0.60
0.59
0.59
0.59
0.58
0.58
0.57
0.57
0.57
0.57
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.57
0.57
0.57
0.58
0.59
0.60
0.63
0.67
0.75
1.00
6,069.48
6,394.58
6,818.28
6,949.54
6,727.23
6,997.57
7,111.90
6,975.91
7,094.48
7,054.71
6,967.50
6,387.38
5,880.42
6,309.03
6,648.02
6,882.78
7,219.58
7,429.60
7,834.29
8,132.71
8,401.06
8,227.02
8,354.68
8,215.15
8,233.98
8,481.00
8,813.42
8,733.50
9,009.18
9,135.49
9,046.62
9,002.83
9,243.38
9,387.23
9,600.17
9,532.93
9,356.17
9,237.02
9,218.34
9,242.82
9,072.49
8,874.41
8,556.37
8,426.27
8,158.47
7,745.83
7,078.31
6,876.17
6,770.09
6,252.49
5,521.92
5,359.27
5,027.23
4,343.35
4,180.27
3,771.06
2,981.69
2,210.22
1,308.69
0.00
13,930.52
14,198.57
14,817.53
14,901.35
14,125.61
14,430.69
14,430.83
13,867.73
13,838.82
13,518.96
13,154.64
12,332.29
11,770.13
11,906.61
11,997.12
12,008.02
12,133.76
12,138.47
12,422.01
12,628.90
12,856.82
12,148.50
12,038.36
11,532.52
11,269.64
11,279.49
11,424.64
10,977.17
11,013.67
10,837.50
10,380.01
10,010.52
9,931.53
9,743.14
9,615.89
9,203.05
8,733.27
8,340.71
8,024.12
7,733.61
7,350.17
6,973.10
6,571.90
6,249.34
5,895.98
5,523.33
5,123.72
4,825.62
4,545.18
4,213.72
3,875.08
3,603.08
3,320.09
3,021.95
2,763.55
2,501.20
2,242.73
2,016.28
1,852.74
1,924.37
INTEREST RATE RISK MANAGEMENT
Figure 3.4
95
Performance Immunization Strategy in Simulations
900
800
Number of Events in 10000 Simulations
700
600
500
400
300
200
100
0
0
100,000
200,000
Dollars
300,000
400,000
96
BASICS OF INTEREST RATE RISK MANAGEMENT
Figure 3.5
Performance Fixed Investment Strategy in Simulations
A. 100% in T−bond
B. 70% in T−Bond
800
1200
Number of Events
Number of Events
1000
600
400
200
800
600
400
200
0
−3,000,000
−1,500,000
Dollars
0
−200,000
0
0
Dollars
C. 30% in T−Bond
D. 100% in Cash
2000
2500
1500
Number of Events
Number of Events
200,000
1000
500
2000
1500
1000
500
0
−1,000,000 0
4,000,000
0
−2,000,000
Dollars
0
6,000,000
Dollars
3.5 plot the results of an analysis similar to the immunization strategy in Figure 3.4,
but in which the investment in long-term T-bond xt is kept fixed at 100% (Panel
A), 70% (Panel B), 30% (Panel C), 0% (Panel D). The figure clearly shows that in
these cases, the bank stands to lose money with some probability. For instance, if
the financial institution was to keep 100% invested in the long term T-bond, it would
stand to lose money about 50% of the time. Instead, the constant strategy of 70%
and 30% in the long term bond stands to lose money about 10% and 40% of the time,
respectively. On the other hand, a 100% investment in cash only (Panel D) also is
not appropriate, as the strategy loses money again about 50% of the time.
3.3.3 Why Does the Immunization Strategy Work?
What is the intuition behind immunization strategies? Why do they work? Think again
about the two extremes: 100% investment in long-term bonds loses money when interest
ASSET-LIABILITY MANAGEMENT
97
Table 3.4 The Duration Mismatch
Commercial Banks
Insurance Companies
Pension Plans
Corporations
Assets
Liabilities
Long-term loans (High D)
Short-Term T-Bonds (Low D)
Medium-Term T-Bonds (Low D)
Long-Term Receivables (High D)
Deposits (Low D)
Long-Term Commitments (High D)
Long-Term Commitments (High D)
Floating Rate Bonds (Low D)
rates go up, because bond prices decline when interest rates increase. Similarly, a 100%
investment in cash loses money when interest rates go down. If the interest rate goes to
zero, for instance, then there is not enough capital to make up the annuity coupon. Clearly,
the safer strategy is in the middle. Indeed, the immunization strategy effectively ensures
that the losses on the cash investment due to declining interest rates are compensated by the
capital gains on the long-term bond. The duration enters the picture here, as it measures
the sensitivity of bond prices to interest rate changes.
3.4
ASSET-LIABILITY MANAGEMENT
Asset liability management is the most classic example of interest rate risk management.
Many financial institutions have a duration mismatch between their assets and their liabilities, as illustrated in Table 3.4. For instance, a commercial bank collects deposits – a
short-term liability whose interest rate changes daily – to make medium- and long-term
loans to other business or households. If the medium- and long-term loans have fixed
coupons, as in fixed rate mortgages for instance, then the duration of the assets is relatively
long, for instance 5 years or more. On the other hand, deposits have a duration close to
zero, as the short term interest rate needs to be adjusted frequently as market conditions
change.
What happens if there is a hike in interest rates?
The analysis in previous sections shows that the value of the assets drop, while the value
of the liabilities does not change. In flow terms, the bank now has to pay a high rate on the
deposits, but still receives a low coupon from its assets. In essence, the bank is in trouble.
The duration analysis in the previous sections can be applied more generally to analyze
the relative potential duration mismatch between assets and liabilities. One important
problem is that financial institutions have very complex asset composition. However, quite
independent of the types of assets, it is possible to compute the duration of the overall
portfolio of assets. Indeed, we can consider the total assets of the firm as a portfolio of
securities (e.g. individual loans, receivables, and so on) and thus use the earlier formula in
Equation 3.8 to compute the duration of assets as a weighted average of the durations of its
components. For instance, if a firm has n individual loans, whose values are A1 , A2 ,...,An
and their durations are DA ,1 , DA ,2 ,..., DA ,n , then we can compute the duration of assets as
n
Duration of assets DA =
wA ,i DA ,i
i=1
where
wA ,i =
Ai
n
i=1
Ai
98
BASICS OF INTEREST RATE RISK MANAGEMENT
Similarly, financial institutions also have very complex liabilities, as they do not finance
their loans only with deposits, but also with longer-term vehicles (e.g., certificates of
deposit), long-term bonds, and, of course, equity. In the same fashion as with assets, the
financial institution can consider its liabilities as a portfolio and compute the duration of
liabilities. Denoting L1 , L2 ,...,Lm the current value of each of its m liabilities (excluding
equity), and DL ,1 , DL ,2 ,...,DL ,m their durations, we obtain
n
Duration of liabilities DL =
wL ,i DL ,i
i=1
where
wL ,i =
Li
n
i=1
Li
The aim of asset - liability management is often taken to minimize the impact that the
variation in the level of interest rates has on the value of equity. Since equity E is given by
total assets (A) minus total liability (L),
E =A−L
we have that duration mismatch occurs whenever DE = 0. Treating equity as a portfolio,
we obtain
A
L
× DA −
× DL
(3.44)
DE =
A−L
A−L
Therefore, DA$ = ADA = LDL = DL$ results in a duration mismatch problem, and
variation of interest rates affect the value of equity.
EXAMPLE 3.11
Consider a hypothetical financial institution mainly engaged in making long-term
loans. The balance sheet of such financial institution may look like the one in Table
3.5. Total assets are around $2.4 billion, with a dollar duration of $19.74 billion. Total
liabilities are $1.8 billion with a dollar duration of only $5 billion. As a consequence,
the market value of equity is $600 million, but with a dollar duration of $14.740
billion. The implication of this mismatch is that a parallel upward shift in interest
rates of 1% generates a decline in assets far greater than in liabilities, implying an
equity decline of $147.4 million. In percentage, this corresponds to a 24%decline in
market value of equity.
To reduce or eliminate this maturity mismatch, the financial firm may alter the
composition of its portfolio. One possibility is to issue long-term debt to increase
the duration of liabilities. Intuitively, if interest rates increase the financial institution
gains from making coupon payments on its long term debt that are below the current
rate. Another far more common possibility is to use derivative securities, such as
swaps, to alter the duration of assets. We explore further this methodology in Chapter
5, after we cover the properties of swaps and other derivative securities.
3.5 SUMMARY
In this chapter we covered the following topics:
EXERCISES
99
Table 3.5 Asset and Liabilities of a Financial Institution
Assets
Item
Cash
S.T. Loans
M.T. Loans
L.T. Loans
Total
Liabilities
Amount
Duration
100
300
500
1500
2400
0
0.8
3
12
Dollar
Duration
0
240
1500
18000
19740
Item
Amount
Duration
Deposits
S.T. Debt
M.T. Debt
L.T. Debt
Total
600
400
400
400
1800
0
0.5
4
8
Equity
600
Dollar
Duration
0
200
1600
3200
5000
14740
1. Duration: The (negative of the) percentage sensitivity of a security to parallel shift in
the term structure of interest rates is known as duration. As an example, the duration
of zero coupon bonds is just their time to maturity.
2. Duration of a portfolio of securities: This can be computed as the weighted average
of durations of the individual securities in the portfolio, where the weights equal the
percentage holdings of the securities.
3. Dollar duration: Unlike duration, the dollar duration measures the (negative of the)
dollar changes in prices due to a parallel shift in the term structure of interest rates.
This can be used for securities or strategies that require a zero investment.
4. Value-at-Risk: VaR is a risk measure that computes the maximum losses a portfolio
can sustain, within a given horizon, with a given probability. For instance, a 95%,
one month VaR provides the maximum loss a portfolio sustains with 95% probability.
5. Expected Shortfall: A risk measure that computes the expected losses on a portfolio,
conditional on these losses being larger than VaR, expected shortfall is a measure
that is better able to deal with tail events than VaR.
6. Immunization: Immunization is a strategy to make a portfolio insensitive to changes
in interest rate.
7. Asset-Liability Management: This is a strategy of choosing the (dollar) duration of
liabilities to match the (dollar) duration of assets. It helps reduce the sensisitivy of
equity to changes in interest rates, and ensures that cash flows received from assets
are sufficient to pay the cash flows from liabilities.
3.6
EXERCISES
1. Today is May 15, 2000, and the current, semi-annually compounded yield curve is
in Table 3.6. Compute the duration for the following securities:
(a) 3-year zero coupon bond
(b) 3 1/4-year coupon bond paying 6% semiannually
(c) 1-year coupon bond paying 4% quarterly
100
BASICS OF INTEREST RATE RISK MANAGEMENT
Table 3.6
Yield Curve on May 15, 2000
Maturity
Yield
Maturity
Yield
Maturity
Yield
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
6.33%
6.49%
6.62%
6.71%
6.79%
6.84%
6.87%
6.88%
6.89%
6.88%
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
6.86%
6.83%
6.80%
6.76%
6.72%
6.67%
6.62%
6.57%
6.51%
6.45%
5.25
5.50
5.75
6.00
6.25
6.50
6.75
7.00
7.25
7.50
6.39%
6.31%
6.24%
6.15%
6.05%
5.94%
5.81%
5.67%
5.50%
5.31%
Notes: Yields are calculated based on data from CRSP (Daily Treasuries).
(d) 6-year floating rate bond with a zero spread, paying semiannually
(e) 3-year floating rate bond with a 35 basis point spread, paid semiannually
(f) 4 1/4 year floating rate bond with 50 basis point spread, paid semiannually
2. An investor is planning a $100 million short-term investment and is going to choose
among two different portfolios. This investor is seriously worried about interest rate
volatility in the market. Compute the duration of the portfolios. Which one is more
adequate for the investor’s objective? Assume today is May 15, 2000, which means
you may use the yield curve presented in Table 3.6
Portfolio A
• 40% invested in 4 1/4-year bonds paying 5% semiannually
• 25% invested in 7-year bonds paying 2.5% semiannually
• 20% invested in 1 3/4-year floating rate bonds with a 30 basis point spread,
paying semiannually
• 10% invested in 1-year zero coupon bonds
• 5% invested in 2-year bonds paying 3% quarterly
Portfolio B
•
•
•
•
40% invested in 7-year bonds paying 10% semiannually
25% invested in 4 1/4-year bonds paying 3% quarterly
20% invested in 90-day zero coupon bonds
10% invested in 2-year floating rate bonds with zero spread, paying semiannually
• 5% invested in 1 1/2 -year bonds paying 6% semiannually
3. Compute the Macaulay and modified duration for the same securities as in Exercise
1.
4. Using the yield curve in Table 3.6, compute the dollar duration for the following
securities:
EXERCISES
101
(a) Long a 5-year coupon bond paying 4% semiannually
(b) Short a 7-year zero coupon bond
(c) Long a 3 1/2-year coupon bond paying 7% quarterly
(d) Long a 2-year zero spread floating rate bond paid semiannually
(e) Short a 2 1/4-year zero spread floating rate bond paid semiannually
(f) Short a 5 1/4-year floating rate bond with a 25 basis point spread paid semiannually
5. The investor in Exercise 2 is still worried about interest rate volatility. Instead of a
duration measure, the investor wants now to know the following:
(a) What is the dollar duration of each portfolio?
(b) What is PV01 for each portfolio?
(c) Does the conclusion arrived at in Exercise 2 stand?
6. Due to a series of unfortunate events, the investor in Exercise 2 just found out that
he must raise $50 million. The investor decides to short the long-term bonds in each
portfolio to raise the $50 million. In other words, for portfolio A the investor would
spend the same on all securities except for the 7-year coupon bonds (paying 2.5%
semiannually) from which the investor will short enough to get to $50 million. For
portfolio B the investor would spend the same on all other securities except for the
7-year coupon bonds (paying 10% semiannually) from which the investor will short
enough to get to $50 million.
(a) How many bonds of each kind does the investor have to short?
(b) What is the new dollar duration of each portfolio?
(c) Does the conclusion arrived at in Exercise 2 stand?
Exercises 7 to 12 use the two yield curves at two moments in time in Table 3.7, and
the following portfolio:
• Long $20 million of a 6-year inverse floaters with the following quarterly
coupon:
Coupon at t = 20% − r4 (t − 0.25)
where r4 (t) denotes the quarterly compounded, 3-month rate.
• Long $20 million of 4-year floating rate bonds with a 45 basis point spread
paying semiannually.
• Short $30 million of a 5-year zero coupon bond.
7. You are standing on February 15, 1994:
(a) What is the total value of the portfolio?
(b) Compute the dollar duration of the portfolio.
8. You are worried about interest rate volatility. You decide to hedge your portfolio
with a 3-year coupon bond paying 4% on a semiannual basis.
102
BASICS OF INTEREST RATE RISK MANAGEMENT
Table 3.7 Two Term Structures of Interest Rates
Maturity
02/15/94
Yield (c.c.)
02/15/94
Z(t, T )
05/13/94
Yield (c.c.)
05/13/94
Z(t, T )
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
5.25
5.50
5.75
6.00
3.53%
3.56%
3.77%
3.82%
3.97%
4.14%
4.23%
4.43%
4.53%
4.57%
4.71%
4.76%
4.89%
4.98%
5.07%
5.13%
5.18%
5.26%
5.31%
5.38%
5.42%
5.43%
5.49%
5.53%
0.9912
0.9824
0.9721
0.9625
0.9516
0.9398
0.9287
0.9151
0.9031
0.8921
0.8786
0.8670
0.8531
0.8400
0.8268
0.8145
0.8023
0.7893
0.7770
0.7641
0.7525
0.7418
0.7293
0.7176
4.13%
4.74%
5.07%
5.19%
5.49%
5.64%
5.89%
6.04%
6.13%
6.23%
6.31%
6.39%
6.42%
6.52%
6.61%
6.66%
6.71%
6.73%
6.77%
6.83%
6.86%
6.89%
6.93%
6.88%
0.9897
0.9766
0.9627
0.9495
0.9337
0.9189
0.9020
0.8862
0.8712
0.8558
0.8406
0.8255
0.8117
0.7959
0.7805
0.7663
0.7519
0.7387
0.7251
0.7106
0.6977
0.6846
0.6713
0.6619
Notes: Yields are calculated based on data from CRSP (Daily Treasuries).
CASE STUDY: THE 1994 BANKRUPTCY OF ORANGE COUNTY
103
(a) How much should you go short/long on this bond in order to make it immune
to interest rate changes?
(b) What is the total value of the portfolio now?
9. Assume that it is now May 13, 1994 and that the yield curve has changed accordingly
(see Table 3.7).
(a) What is the value of the unhedged portfolio now?
(b) What is the value of the hedged portfolio?
(c) Is the value the same? Did the immunization strategy work? How do you
know that changes in value are not a product of coupon payments made over
the period?
10. Instead of assuming that the change took place 6 months later, assume that the change
in the yield curve occurred an instant after February 15, 1994.
(a) What is the value of the unhedged portfolio?
(b) What is the value of the hedged portfolio?
11. Now use the February 15, 1994 yield curve to price the stream of cash flows on May
13, 1994.
(a) What is the value of the unhedged portfolio?
(b) What is the value of the hedged portfolio?
12. From the answers to the Exercises 7 - 11, answer the following:
(a) What is the change in value in the portfolio due to the change in time only,
without change in interest rates?
(b) Is this difference a loss?
(c) Once we have adjusted for paid coupons, what is the change in value of the
portfolio due to interest rate movements?
3.7
CASE STUDY: THE 1994 BANKRUPTCY OF ORANGE COUNTY
As discussed in Section 3.1.2, in 1994 Orange County lost $1.6 billion out of a portfolio
of $7.5 billion in assets as a result of an unexpected increase in interest rates, from 3% to
5.7%.6 Figure 3.6 shows the sudden steep increase in the level of interest rates in 1994.
The lessons learned in this chapter will help us understand what type of exposure Orange
County had in its portfolio that could lead to a loss of this magnitude.
6 This
section and the next are based on publicly available information and they are only meant to illustrate the
concepts introduced in this chapter. No claim of wrongdoing by any party is made here. Descriptive material
is from the case study ERISK: Orange County, downloaded from http://www.erisk.com/Learning/CaseStudies/
OrangeCounty.asp.
104
BASICS OF INTEREST RATE RISK MANAGEMENT
Figure 3.6
The Level of Interest Rates, 1992 - 1994
7
6.5
Interest Rate (%)
6
5.5
5
4.5
4
3.5
1992
1992.5
1993
1993.5
1994
1994.5
1995
Data Source: CRSP.
3.7.1 Benchmark: What if Orange County was Invested in Zero Coupon
Bonds Only?
A useful starting point is to suppose that the Orange County portfolio was invested only
in zero coupon bonds and then find the maturity of these bonds necessary to bring a loss
of $1.6 billion. We can do this by using the concept of duration introduced in Section 3.2.
Recall that duration is defined as the (negative of the) sensitivity of a security, or a portfolio,
to parallel shifts in the term structure of interest rates. More specifically, Equation 3.1
defines duration as
1 dP
(3.45)
Duration = DP = −
P dr
We know that the portfolio value before the hike in interest rates was approximately $7.5
billion, and thus P = 7.5. In addition, the change in the level of interest rates was
dr = 6.7% − 4% = 0.027. Finally, the loss was dP = −1.6. Substituting into Equation
3.45 we find
1 −1.6
Duration = −
= 7.90
(3.46)
7.5 .027
That is, if Orange County’s portfolio was invested only in Treasury securities, given the
ex-post losses, we would gather that the duration of this portfolio should have been around
7.90. From Equations 3.5 and 3.6, the duration of a zero coupon bond is equal to its time
to maturity. Thus, Orange County would have been hit by the same type of losses as if all
of its portfolio was invested in zero coupon bonds with 7.9 years to maturity.
CASE STUDY: THE 1994 BANKRUPTCY OF ORANGE COUNTY
3.7.2
105
The Risk in Leverage
Orange County’s portfolio, however, was not only invested in Treasury securities. By using
the repo market, Orange County effectively levered up the portfolio position to $20.5 billion
(see Chapter 1 to review the repurchase agreements contract.) Essentially, the Treasurer of
Orange County could pledge the portfolio’s highly liquid Treasury securities as collateral
in a repo transaction so as to obtain other funds to invest further in Treasuries or other
securities.
For now, we only consider the effect of leverage. First, the duration of the levered
portfolio now has to be computed using the assets and liability formula in Equation 3.44,
that is
(3.47)
Duration leveraged portfolio = wA × DA + wL × DL
where wA = A/(A − L) = 20.5/7.5 and wL = −L/(A − L). The liability is given by the
repo transactions, which are financed at the overnight rate. The duration of liabilities, thus,
is approximately zero. That is, DL ≈ 0. The duration of assets, in contrast, is given by
DA = −
1 −1.6
= 2.89
20.5 .027
(3.48)
Of course, wA × DA = 7.90 as before. The point of this calculation, however, is that the
portfolio of Orange County may well have been invested only in short-term zero coupon
bonds with maturity 2.89. Although such an investment per se appears very safe, the
presence of the large leveraged position generates a much higher duration of the leveraged
portfolio itself, which could lead to the $1.6 billion losses when the interest rates moved
by 2.7%, as they did in 1994.
3.7.3
The Risk in Inverse Floaters
Although the main reason for having a record-breaking loss in the Orange County portfolio
was leverage, the trigger was the change in interest rates and its effect on inverse floaters.
Recall from Chapter 2 that inverse floaters have a coupon that moves inversely to shortterm floating rates. This implies that when interest rates go up, the price of inverse floaters
receive a negative shock from two channels:
1. The discount channel: If interest rates go up, prices of zero coupon bonds fall as
future cash flows are worth less in today’s money.
2. The cash flow channel: If interest rates go up, the actual cash flow is reduced because
coupon payments move inversely to interest rates.
The sensitivity of inverse floaters to interest rates can be calculated through the concept of
duration. How can we compute the duration of an inverse floater? Here, we must remember
that an inverse floater is given by a portfolio of more basic securities, of which we can
compute the duration easily.
In particular, recall from Chapter 2, Equation 2.43, that we can write the price of a
(plain vanilla) inverse floater with maturity T with annual payments and coupon c(t) =
c − r1 (t − 1) as
Price inverse floater PI F (0, T ) = Pz (0, T ) + Pc (0, T ) − PF R (0, T )
(3.49)
106
BASICS OF INTEREST RATE RISK MANAGEMENT
Table 3.8
The Duration of the 15% Fixed Rate Bond
Date
Cash Flow
Discounted
Cash Flow
Weight
w
T
w∗T
12/31/1994
12/31/1995
12/31/1996
0.15
0.15
1.15
0.1446
0.1379
1.0057
0.1123
0.1070
0.7807
1
2
3
0.1123
0.2141
2.3421
Total Value
1.2884
Duration:
2.6685
where Pz (0, T ) is the price of a zero coupon bond, Pc (0, T ) is the price of a c coupon bond,
and PF R (0, T ) is the price of a floating rate bond, all of them with maturity T . Thus, we
can compute the duration of the inverse floater by applying the formula for the duration of
a portfolio, namely, Equation 3.10.
We consider here the simple case discussed in Chapter 2, c = 15%, T = 3 and coupon
payments are annual. In this case, we obtained PI F (0, 3) = $116.28, Pz (0, 3) = 87.45,
Pc (0, 3) = 128.83, and PF R (0, 3) = 100. The duration of the inverse floater can be
computed then as
DI n v er se = wZ er o × Dz er o + wF ixed × DF ixed + wF loatin g × DF loatin g
(3.50)
where wZ er o = Pz (0, 3)/PI F (0, 3) = 0.7521, wF ixed = Pc (0, 3)/PI F (0, 3) = 1.1079,
and wF loatin g = −PF R (0, 3)/PI F (0, 3) − .8600 are the weights. The duration of a zero
coupon bond equals its time to maturity, thus Dz er o = 3. The duration of a floating rate
bond with annual coupons is equal to the time of the first coupon at reset dates. Thus,
DF loatin g = 1. The only term left to calculate is the duration of the fixed rate bond DF ixed .
Table 3.8 performs the computation, obtaining DF ixed = 2.6684.
We can substitute everything into Equation 3.50, to find
DI n v er se
= wZ er o × Dz er o + wF ixed × DF ixed + wF loatin g × DF loatin g
=
0.7521 × 3 + 1.1079 × 2.6685 − .8600 × 1
=
4.35
The duration of the 3-year inverse floater is 4.35. It is important to note that the duration
is higher than the inverse floater maturity (3 years). In this sense, the notion of “duration”
as a temporal average of cash flows plays no role here, as we are interpreting the duration
as the sensitivity of the security’s price to changes in interest rates. Depending on how cash
flows move with interest rates, this sensitivity can be larger or smaller than the maturity of
the security itself.
3.7.4 The Risk in Leveraged Inverse Floaters
Recall that a leveraged inverse floater has a coupon that moves (inversely) to interest rates
by more than one-to-one. For instance, the leveraged inverse floater discussed in Section
2.8.4 of Chapter 2 has a coupon
c(t) = 25% − 2 × r1 (t − 1)
(3.51)
CASE STUDY: THE 1994 BANKRUPTCY OF ORANGE COUNTY
107
Table 3.9 The Duration of the Leverage Inverse Floater
Security
Value
Weight w
Duration D
D∗w
2 × Pz (3)
Pc (3)
−2 × PF R (3)
174.91
156.41
-200.00
1.3320
1.1911
-1.5231
3.00
2.5448
1.00
3.9959
3.0311
-1.5231
Total Value:
103.78
Duration:
5.5040
Recall also from Chapter 2 that the price of the leveraged inverse floater can be computed
as:7
Price leveraged inverse floaterPL I F (0, T ) = 2 × Pz (0, T ) + Pc (0, T ) − 2 × PF R (0, T )
(3.52)
To compute the duration of the leveraged inverse floater we need to compute the duration of
the fixed-coupon bond. Using the same steps as in Table 3.8 but with coupon rate c = 25%
we find that the duration of the coupon bond in this case is Dc = 2.5448. Given this
information, we can now compute the duration of the leveraged inverse floater. Table 3.9
contains the calculations. The 3-year leveraged inverse floater has a duration of 5.5040,
almost twice its maturity. This security is very sensitive to changes in interest rates, indeed.
3.7.5
What Can We Infer about the Orange County Portfolio?
With these data we can get a sense of the composition of the Orange County portfolio. It
appears that the portfolio had about $2.8 billion in “inverse floaters [...], index amortizing
notes, and collateralized mortgage obbligations.”8 For simplicity, we assume that $2.8
billion was invested only in leveraged inverse floaters. Assuming the remaining part of the
portfolio was invested in safe Treasury securities, what should have the duration of this
additional investment been?
Let x = 2.8/20.5 = 0.1366 be the fraction of total assets invested in leveraged inverse
floaters. Then, we know that
Duration of assets = x×Duration of leveraged inverse floater+(1−x)×Duration of T-bills
(3.53)
From x = 0.1366, the duration of leveraged inverse floaters (= 5.5040) and the duration
of assets (= 2.89)
Duration of T-bills =
2.89 − 0.1366 × 5.5040
= 2.4764
1 − 0.1366
(3.54)
That is, the $20.5 billion Orange County portfolio could well have been mainly invested in
short-term Treasury bonds (with duration of only 2.4764). Yet, the large leverage and the
very high duration of leveraged inverse floaters may still have produced large losses as the
interest rate increased.
7 We
8 See
are still making the simplifying assumption that we know that c(t) > 0 for sure, i.e. that r 1 (t) < 25%/2.
ERisk Case, Orange County (2001), page 2.
108
BASICS OF INTEREST RATE RISK MANAGEMENT
3.7.6 Conclusion
In conclusion, this case illustrates the risk embedded in fixed income securities, and, in
particular, in leveraged positions. Structured securities, such us leveraged inverse floaters,
contain additional risks that the risk manager must be aware of. In particular, this case
emphasizes that even if the average maturity of the instruments may be very low, the risk of
such securities or portfolio may be very high. In this sense, the interpretation of duration
as the weighted average of cash flow payments is strongly misleading. As illustrated in
the case, the Orange County portfolio could well have been mainly invested in short-term
Treasuries and leveraged inverse floaters. Yet, this portfolio still has a large sensitivity to
interest rates, and therefore it is very risky.
3.8 CASE ANALYSIS: THE EX-ANTE RISK IN ORANGE COUNTY’S
PORTFOLIO
In hindsight it seems that Orange County’s investment strategy paved the way for its
own disaster, but any reasonable assessment must be made using ex ante information. In
particular, was there anything that ex ante could have warned Orange County’s Treasurer
and its creditors regarding the potential risk that the portfolio was bearing? We can answer
this question by using the concepts of Value-at-Risk and expected shortfall introduced
in Sections 3.2.8 and 3.2.9, respectively. We compute these risk measures under both the
historical distribution and normal distribution approach by making use of all the information
available up to January 1994.9
1. Historical Distribution Approach. We can use the past changes in the level of
interest rates dr as a basis to evaluate the potential changes in a portfolio value dP .
Panel A of Figure 3.7 shows the historical changes in the average level of interest
rates at the monthly frequency. Panel B makes a histogram of these changes, that is,
describes the frequency of each possible change. As can be seen, large increases and
decreases are not very likely, but they do occur occasionally. We can now multiply
each of these changes dr observed in the plot by −DP × P to obtain the variation in
dP . Figure 3.8 plots the histogram of the changes in the portfolio (i.e. the portfolio
profit and loss, or P&L). Given this distribution, we can compute the maximum loss
that can occur with 99% probability. We can start from the left-hand side of the
distribution, and move right until we count 1% of the observations. That number
is the 99% monthly VaR computed using the historical distribution approach. In
this case, we find it equal to $715 million. That is, there is only 1% probability
that Orange County portfolio could lose more than $715 million in one month. The
corresponding expected shortfall, obtained by averaging all of the portfolio losses
that are lower than $715 million, turns out to be $990 million. That is, the expected
monthly loss of the Orange County portfolio in case of an extreme event is $990
million.
2. Normal Distribution Approach. We can also use some assumption about the
distribution of interest rates. For instance, if dr is normally distributed, so is dP .
9 It
should be mentioned that by this date Value-at-Risk and expected shortfall were not yet been introduced as
risk measures, and therefore Orange County’s Treasurer could have not done the following calculations.
CASE ANALYSIS: THE EX-ANTE RISK IN ORANGE COUNTY’S PORTFOLIO
109
From the data in the top panel of Figure 3.7 we can compute the historical mean and
standard deviation of dr, and thus obtain the mean and standard deviation of dP . In
particular, we find
Mean(dr) = μ
dr = 4.71E −05 ; Std(dr) = σ
dr = 0.00432;
which implies
Mean(dP ) = −DP × P × μ
dr = −0.0028; Std(dP ) = DP × P × σ
dr = 0.2563
Figure 3.8 also reports the normal density with mean μP = −.0028 and standard
deviation σ dP = 0.2563. In this case, the 99% maximum loss can be computed from
the properties of the normal distribution, resulting in VaR = −(μP −2.326×σ dP ) =
$598 million. This number is smaller than the one obtained under the historical
distribution approach, because of the fat-tailed distribution of the portfolio P&L,
as shown in Figure 3.8: Extreme realizations are more likely under the historical
distribution than under the normal distribution. Indeed, from Fact 3.7 the 99%
expected shortfall in the case of the normal distribution is only $680 million, which
is much smaller than the $990 million expected shortfall obtained under the historical
distribution approach.
The VaR numbers computed above are relatively small compared to the ex-post $1.6
billion loss. It is important to realize, though, that the VaR so computed is a monthly figure,
while Orange County losses accrued over a six-month period. How can we compute a
6-month VaR? Using the normal distribution approach and assuming that monthly changes
in interest rates are independent and identically distributed – a strong assumption as there is
some predictability in yields, as discussed in Chapter 7 – the annualization can be√performed
by multiplying the mean μP by 6 and the monthly standard deviation σ P by 6. In this
case we obtain a 99% 6-month VaR equal to $1.48 billion, close to the actual loss suffered
by Orange County.
3.8.1
The Importance of the Sampling Period
The VaR calculation is very sensitive to the sample used in the calculation. In fact, the top
panel of Figure 3.7 shows that the volatility of the level of interest rates had been relatively
low in the decade before 1994. The large estimate of the monthly standard deviation of
interest rates σ dr = 0.00432 is mainly due to the large volatility in the 1970s and beginning
of the 1980s. If we restrict the sample to compute the standard deviation of interest rate
changes dr to the more recent period, such as five years, we find the much smaller standard
deviation σ dr = 0.0028. The 6-month, 99% VaR in this case is $668 million, a large
number, but much smaller than the actual ex-post losses.
What sample period is more relevant? The large shift up in the interest rate was probably
unexpected. However, it is a common mistake to confuse a low volatility period as a safe
period. History shows time and again that low volatility periods are followed by high
volatility periods. Thus, reliance on the recent past would miss the probability that in fact
volatility will go up, and with it, the risk of large losses. The use of a longer sample that
takes into account such facts is therefore more conservative for the risk manager.
110
BASICS OF INTEREST RATE RISK MANAGEMENT
Figure 3.7
The Monthly Changes in the Average Level of Interest Rates
Change in Interest Rate (%)
Panel A. Monthly Variation
3
2
1
0
−1
−2
−3
1955
1960
1965
1970
1975
1980
1985
1990
1995
Panel B. Histogram
120
Number of Events
100
80
60
40
20
0
−4
−3
−2
−1
0
1
Change in the Level of Interest Rates (%)
2
3
Data Source: CRSP.
3.8.2 Conclusion
The ex-ante measurement of risk is difficult and full of potential pitfalls. In the previous
section, we computed several numbers in the attempt to measure the risk embedded in
the Orange County portfolio. The numbers we computed vary greatly depending on (a)
the type of model (e.g., normal versus historical); (b) the horizon (one month versus six
months); (c) the sample used (last five years versus longer sample); (d) the type of risk
measure (VaR versus Expected Shortfall).
The natural question is then the following: Which one of these measures is best?
Unfortunately, this is hard to tell. While it sounds intuitive that we should always use the
most conservative measure of risk, i.e. the one implying the largest possible losses, there
are good reasons also to not be too conservative, as a portfolio manager who is overly
conservative may miss important profit opportunities. The main goal of this case is to show
that the measurement itself of risk is difficult, and so the portfolio manager should always
be suspicious of any risk measure, and always ask how such a measure was calculated. The
major risk for a portfolio manager is in fact to rely too much on these measures of risk,
APPENDIX: EXPECTED SHORTFALL UNDER THE NORMAL DISTRIBUTION
Figure 3.8
111
The Distribution of the Monthly P&L of the Orange County Portfolio
3.5
3
Normal Distribution
Approach
99% VaR = $598 mil
Probability Density
2.5
2
Historical Distribution
Approach
99% VaR = $715 mil
1.5
1
0.5
0
−2
−1.5
−1
−0.5
0
Billions of Dollars
0.5
1
1.5
2
forgetting that they are fragile, in the sense that they greatly depend on the way they are
computed.
3.9
APPENDIX: EXPECTED SHORTFALL UNDER THE NORMAL
DISTRIBUTION
In this appendix we derive the formula in Equation 3.34 for the expected shortfall under
the normal distribution case. Let dP denote the P&L of the portfolio, so that the loss
is LT = −dP . Clearly, LT > V aR when dP < −V aR. Let z be the “quantile”
corresponding to the VaR. For instance, z = 1.645 for the 95% VaR, and recall from
Equation 3.30, dP ∼ N (μP , σ P ). We then have
E [LT |LT > V aR] = −E [dP |dP < −V aR] = −E [dP |dP < (μP − σ P × z)]
dP − μP
dP − μP
= − μP + σ P × E
|
< −z)
σP
σP
(3.55)
112
BASICS OF INTEREST RATE RISK MANAGEMENT
The quantity X =
dP −μ P
σP
is a standardized normal distribution. Since
−z
E[X|X < −z] = ∞−z
∞
xf (x)dx
f (x)dx
=
−f (−z)
N (−z)
the formula in Equation 3.34 follows from substituting this latter expression into Equation
3.55.
CHAPTER 4
BASIC REFINEMENTS IN INTEREST RATE
RISK MANAGEMENT
In this chapter we review some basic refinements in interest rate risk management. The
concept of duration discussed in Chapter 3 is a good first approximation to measure the
risk embedded in fixed income instruments. However, it is possible to improve upon it,
and this is accomplished in two ways: First, we realize that the relation between a bond
price and the interest rate is not linear, which is the implicit assumption in the duration
approximation. Second, we also realize that the term structure of interest rates does not
move in a parallel fashion, which is a second important assumption of the duration concept.
By generalizing the risk model along these two dimensions we are able to obtain more
precise measures of risk, as well as improve upon risk management practices.
4.1
CONVEXITY
The relation between bond prices and interest rates is not linear. Assume for instance a
flat term structure of interest rates at rates r = .01, r = .02,...,r = .15. Figure 4.1 plots
the values of 5-, 10-, 20- and 30-year zero coupon bonds against these interest rates. We
see that as the interest rate r increases, the zero coupon bonds decrease, and they become
flatter and flatter as the interest rate becomes higher and higher. This pattern is especially
true for long-dated zero coupon bonds.
This observation has an impact on the practice of interest rate risk management. In
Chapter 3 we explored the notion of duration, that is, the (negative of the) percentage
change in a security price due to a small parallel shift in the term structure of interest rates.
113
114
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
Figure 4.1
Zero Coupon Bond Prices versus Interest Rates
100
5 year
10 year
20 year
30 year
90
80
Zero Coupon Bond Pz(r,t;T)
70
60
50
40
30
20
10
0
0
5
10
15
Interest Rate r (%)
As Figure 4.2 shows, though, if the change in interest rates is substantial, the approximation
that uses the notion of duration is not very good. In the figure, the approximation of the
change in the bond price implemented through the concept of duration follows the straight
line. Instead, the true bond price follows the curved line. Clearly, for small changes in
interest rate, the straight line and the curved line are very close to each other. However,
for large changes, the approximation is relatively poor. For instance, for this 20-year
zero coupon bond, a change of interest rate from 5% to 2.5% implies an increase in the
bond price from $36.79 to $60.65. In contrast, because the duration of a zero coupon
bond equals its maturity D = 20, the duration approximation implies an increase to
$36.79 − D × P × dr = $36.79 + 20 × $36.79 × 2.5% = $55.18, a much smaller value.
To obtain a more accurate measure of the impact that a change in the term structure has
on bond prices, we must take into account the convexity of the bond with respect to the
interest rate. The convexity can be measured by referring to the notion of second derivative.
In a nutshell, while the first derivative measures the slope of a function (see Definition 3.3
in Chapter 3), the second derivative measures the curvature of a function.
Definition 4.1 The second derivative of a function F (x), denoted by
derivative of the first derivative
2
d F (x)
=
d r2
d
d 2 F (x)
d r2 ,
is the first
d F (x)
d r
dr
In particular, let F (x) = A×eax be a function of x. From Definition 3.3, the first derivative
of F (x) is dF
dx = aF (x), still a function of x. Then, the second derivative of F (x) with
CONVEXITY
Figure 4.2
115
Bond Price Approximation with Duration
100
90
80
Zero Coupon Bond Pz(r,t;T)
70
Accurate Approximation
60
50
40
30
20
10
Non Accurate Approximation
0
0
5
10
15
Interest Rate r (%)
respect to x, is given by
Second derivative of F (x) with respect to x =
d2 F
= A × a2 × eax = a2 × F (x)
dx2
What does the second derivative of a function F (x) measure? While the first derivative
of a function F (x) measures the slope of the function at a point x, the second derivative
measures the curvature of the function in the same point x. The curvature of the function
is effectively measured by the change in slope, as illustrated in Figure 4.3. If the change
in slope is zero, then the function F (x) is a straight line, and thus there is no curvature. In
this case, the second derivative is in fact zero.
Putting together the notion of second derivative in Figure 4.3 with the problem of
duration illustrated in Figure 4.2, we see that knowledge of the second derivative of the
bond price function can help to increase the precision of the approximation of bond price
changes due to changes in the interest rate.
Definition 4.2 The convexity of a security with price P measures the percentage change
in the price of the security due to the curvature of the price with respect to the interest rate
r. Formally
Convexity = C =
1 d2 P
P d r2
(4.1)
116
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
Figure 4.3 Second Derivative
100
90
Second Derivative = Change in Slope
80
Zero Coupon Bond Pz(r,t;T)
70
60
50
40
30
20
10
0
0
5
10
15
Interest Rate r (%)
Once we compute the convexity (Equation 4.1) of a security, we can compute a better
approximation of the impact of changes of interest rates on bond prices. In fact, putting
together duration and convexity, we find the following:
Fact 4.1 An approximation of the percentage impact of interest rates on the price of a
security is given by
1
dP
= −D × dr + × C × dr2
(4.2)
P
2
In other words, the convexity term C augments the precision of the approximation made
by the duration term, but it does not substitute for it. We provide some examples below.1
Figure 4.4 depicts the benefits from using both duration and convexity to approximate
the impact of (large) interest rate changes on bond prices. While the straight, dotted line
represents the approximation using only duration, also shown in Figure 4.2, the dashed line
represents the approximation using duration and convexity.
4.1.1 The Convexity of Zero Coupon Bonds
As in the case of duration, convexity can be computed from first principles for coupon
bearing bonds. We start from zero coupon bonds, recalling that the price of a zero coupon
1 Equation
4.2 stems directly from the application of a second-order Taylor expansion to the price of the security,
a result that also shows the need to multiply by 1/2 the convexity term C .
CONVEXITY
Figure 4.4
117
Duration plus Convexity Approximation
100
90
Duration plus Convexity
Approximation
80
Zero Coupon Bond Pz(r,t;T)
70
60
50
40
30
20
10
Duration Approximation
0
0
5
10
15
Interest Rate r (%)
bond is given by
Pz (r, t; T ) = 100 × e−r ×(T −t)
The first derivative of the bond with respect to r is dPz /dr = −(T − t) × Pz (r, t; T ). Thus,
the convexity of the zero coupon bond is
Cz
=
=
=
1
d2 Pz
×
Pz
d r2
1
× {(T − t)2 × Pz (r, t; T )}
Pz
(T − t)2 .
(4.3)
(4.4)
(4.5)
EXAMPLE 4.1
In Figure 4.2 we consider the impact that a 2.5% decline in the interest rate has on
a 20-year zero coupon bond. We discovered that the price changes from $36.79 to
$60.65, while the duration approximation only yields a change to $55.18, a much
smaller value. Convexity helps close the gap. In fact, we have Cz = 202 = 400.
Thus, the percentage change in price due to a 2.5% decline in interest rate is
1
dP
= 20 × 0.025 + × 400 × (0.025)2 = 0.6250
P
2
The approximate price of the bond after the decline is then Pz (r, t; T ) + dP =
$36.76 + 0.625 × $36.76 = $59.78, much closer to the actual value of $60.65.
Figure 4.4 plots this case, in fact.
118
4.1.2
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
The Convexity of a Portfolio of Securities
Like duration, the convexity of a portfolio of securities is equal to the weighted average
of the convexities of the individual securities, where the weights correspond to the relative
value of each asset in the portfolio. The derivation of this formula is identical to the one for
the duration of a portfolio in Section 3.2.2, and we therefore omit it here. The final result
is the following:
Fact 4.2 Let Ni , i = 1, ..., n be the units of securities 1,..,n in a portfolio, and let Pi be
n
their prices. The value of the portfolio is W = i=1 Ni Pi . Let Ci be the convexity of
security i. Then
n
Convexity of portfolio
= CW =
wi Ci
(4.6)
i=1
where
wi =
Ni × P i
W
(4.7)
4.1.3 The Convexity of a Coupon Bond
As an application of the convexity formula in Equation 4.6, we derive the convexity of a
coupon bond. Recalling that a coupon bond can be considered as a portfolio of zero coupon
bonds, we obtain:
Fact 4.3 The convexity of a coupon bond with maturity T , coupon c, price Pc (t, T ) and
n payment times T1 , ... , Tn = T is given by
n
C
=
wi × Cz ,i
(4.8)
i= 1
where Cz ,i = (Ti − t)2 and
wi
=
wn
=
c/2 × Pz (t, Ti )
for i = 1, .., n − 1,
Pc (t, T )
(1 + c/2) × Pz (t, Tn )
.
Pc (t, T )
(4.9)
(4.10)
Substituting, we obtain the convexity formula
C
=
1
×
Pc (t, T )
n −1
i=1
c
c
× Pz (t, Ti ) × (Ti − t)2 + 1 +
× Pz (t, Tn ) × (Tn − t)2
2
2
CONVEXITY
Table 4.1
119
Duration and Convexity Computation for 10-year, 5% Coupon Bond
Period
i
Time
Ti
Cash Flow
CF
Discount
Z(0, Ti )
Discounted
Cash Flow
CF ×Z(0, T 0)
Weight
wi
Weight×Ti
w i × Ti
Weight×Ti2
wi × Ti2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
102.5
0.9778
0.9560
0.9347
0.9139
0.8936
0.8737
0.8543
0.8353
0.8167
0.7985
0.7808
0.7634
0.7464
0.7298
0.7136
0.6977
0.6822
0.6670
0.6521
0.6376
2.44
2.39
2.34
2.28
2.23
2.18
2.14
2.09
2.04
2.00
1.95
1.91
1.87
1.82
1.78
1.74
1.71
1.67
1.63
65.36
0.024
0.023
0.023
0.022
0.022
0.021
0.021
0.020
0.020
0.019
0.019
0.018
0.018
0.018
0.017
0.017
0.016
0.016
0.016
0.631
0.0118
0.0231
0.0338
0.0441
0.0539
0.0633
0.0722
0.0806
0.0887
0.0964
0.1036
0.1106
0.1171
0.1233
0.1292
0.1347
0.1400
0.1449
0.1495
6.3101
0.0059
0.0231
0.0508
0.0882
0.1348
0.1898
0.2526
0.3226
0.3992
0.4818
0.5701
0.6633
0.7612
0.8631
0.9688
1.0778
1.1896
1.3040
1.4206
63.1010
D = 8.0309
C = 73.8682
P = 103.58
EXAMPLE 4.2
A corporation buys $100 million (par value) of a 10-year coupon bond that pays a
5% semi-annual coupon. Assume that the term structure of the interest rates is flat
at the continuously compounded rate of 4.5%. Table 4.1 shows that the price of the
bond is $103.58, implying a position in the bond of $103.50 million and a duration
of 8.03. This duration value implies that an increase of the yield by 1%, from 4.5%
to 5.5%, entails an approximate loss of 8%, as
dP
≈ −D × 0.01 = −0.0803 = −8%
P
Table 4.1 shows that the convexity of the position is 73.87. Adding convexity
instead entails an approximate loss of 7.66%, computed as follows:
dP
1
≈ −D × 0.01 + × C × (0.01)2 = −0.07662 = −7.66%
P
2
We can verify that the latter approximate variation is closer to the real change by
simply recomputing the price of the bond when the term structure is flat at 5.5%. A
120
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
computation analogous to the one in Table 4.1 shows that the price declines to $95.63
from $103.58, implying a loss equal to
dP
$95.63 − $103.58
=
= −7.67%
P
$103.58
This is indeed very close to the calculation made using the convexity adjustment.
4.1.4
Positive Convexity: Good News for Average Returns
Predicting the short-run changes in the level of interest rates is extremely hard. When the
level of interest rates is low, we can expect it to increase in the long term future, and likewise
when the level of interest rates is very high we can expect it to decrease. But predicting
the change in the level of interest rates day by day is extremely difficult. Suppose we have
invested $100 million in a 20-year zero coupon bond. Given a change in the level of interest
rates equal to dr, the duration computation allows us to compute the expected return, by
using the formula
dP
= −D × dr = −20 × dr
(4.11)
P
Thus, if the level of interest increases by one basis point (dr = .01%) we stand to lose
0.2% = 20 × .01% of our investment, that is $200, 000. Similarly, if the level of interest
rate decreases by one basis point (dr = −.01%), we stand to gain $200, 000. Can we
forecast how much money we can make between today and tomorrow? The answer is that
if we cannot forecast the change in interest rate dr we cannot forecast the return itself. We
will use the notation E[dr] to indicate the expected change in the level of interest rate, and
E[dP/P ] to represent the expected return from the investment. Since we just said that
E[dr] = 0, Equation 4.11 implies
dP
E
P
= −20 × E[dr] = 0
What if we consider the convexity term? Using Equation 4.2 and recalling that for a 20-year
zero coupon bond the convexity is C = 400, we find
dP
1
(4.12)
= −20 × E[dr] + × 400 × E[dr2 ]
E
P
2
What is E[dr2 ]? In statistics, when E[dr] = 0 this quantity corresponds to the variance of
the change in level of interest rates. That is, it measures the size of the daily fluctuations
in interest rates. The key insight is that although we do not know the direction in which
the interest rate will move between today and tomorrow, it is extremely likely it will move.
That is, the variance of interest rates is positive, E[dr2 ] > 0. Figure 4.5 plots the daily
changes in the level of interest rates over the past three decades: It appears clear that every
day the level of interest rates moves.2
2 We
computed daily spot curves using the extended Nelson Siegel model based on data from CRSP (Daily
Treasuries) ¤2009 Center for Research in Security Prices (CRSP), The University of Chicago Booth School of
Business.
CONVEXITY
121
Figure 4.5 Daily Changes in the Level of Interest Rates
0.8
0.6
Change in Interest Rate (%)
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
1975
1980
1985
1990
1995
2000
2005
Data Source: CRSP.
Why is this significant to the return on the investment in the 20-year bond? The reason
is that given the variance of interest rates of about E[dr2 ] = 5.5351 × 10−007 we find
dP
1
E
(4.13)
= −20 × 0 + × 400 × E[dr2 ] = 1.11 × 10−04 > 0
P
2
Although this number seems extremely small, it is a daily expected return. To better gauge
the magnitude, we can annualize the expected return by multiplying this number by 252,
the number of trading days in a typical year. In this case, we find
Annualized expected return from convexity = 1.11 × 10−04 × 252 = 2.79%
Similarly, considering the $100 million investment, convexity yields a daily dollar return
of $11, 070 = 1.11 × 10−04 × 100 million.
Is this positive average return stemming from the fluctuation of the level of interest
rates a “free lunch”? No: As we shall see more formally in later chapters, this positive
average return on the investment is counterbalanced by a lower yield to maturity of the
bond, everything else equal. Indeed, no arbitrage conditions entail that a precise relation
must exist between the convexity of the bond and its yield to maturity.
4.1.5
A Common Pitfall
A common pitfall is to think of “convexity” as the “change in duration.” As the definition
above shows, it is not. As a simple example, the duration of a zero coupon bond is constant
122
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
independent of the interest rate, as it is equal to its time to maturity. Yet, as shown earlier,
the convexity of a zero coupon bond is most definitely not zero.3
The correct statement is instead that the curvature of a function, represented by its
second derivative, is indeed equal to the change in slope of the function, given by the first
derivative. Adopting the terminology used in Chapter 3, we can call dollar convexity the
quantity
Dollar convexity = C $ =
d2 P
d r2
(4.14)
In this case, indeed, the dollar convexity is related to the change in dollar duration, defined
in Section 3.2.7. In particular, by the definition of second derivative, the dollar convexity
equals the change of the negative of the dollar duration. Note that the dollar convexity of
a zero coupon bond is not constant, as it equals (T − t) × Pz (r, t; T ) and the zero coupon
bond does depend on the interest rate.
4.1.6 Convexity and Risk Management
Taking into account convexity improves the performance of interest rate risk management
practices, especially in an environment with substantial shifts in the yield curve. To motivate
the convexity hedging strategy, consider again Example 4.2.
EXAMPLE 4.3
Suppose that the corporation in Example 4.2 is worried about the losses that its
portfolio may suffer from an upward shift in the term structure of interest rates. Let
us consider first a duration hedging strategy. For simplicity, let us assume that the
corporation decides to enter into a position of k 10-year zero coupon bond Pz (0, T )
to hedge away the interest rate risk. From Table 4.1, the value of this zero coupon
bond is Pz (0, T ) = 100 × Z(0, T ) = $63.76. Its duration, of course, is Dz = 10.
Denoting again by P the value of the bond (P = $103.58 from Table 4.1), what is
the position k in the zero coupon bond such that the portfolio V = P + k × Pz (0, T )
is insensitive to a parallel shift in the yield curve? From Chapter 3, we must have
dV = 0, which implies
k=−
8.03 × 103.58
D×P
=
= −1.3045
Dz × Pz (0, T )
10 × 63.76
(4.15)
That is, to hedge against a parallel shift in the term structure, the corporation must
short 1.3045 units of a 10-year zero coupon bond. We assume that the corporation
achieves this short position through the repo market (see Chapter 1), that is, it borrows
the zero coupon bond from a repo dealer, sells it to the market for 1.3045×Pz (0, T ) =
3I
cannot resist mentioning the anecdote that made me write this subsection: In my years at Chicago Booth,
several times my students were asked the following questions during job interviews: “What is the duration of
a 20-year zero coupon bond?” My students would reply “20”. The interviewer would then ask: “What is the
convexity of the 20-year zero coupon bond?” My students (always well prepared) would answer “400” (= 20 2 ).
The interviewer would then say “Ah! I caught you! You see, the duration of the zero coupon bond is independent
of interest rates, therefore the convexity of the zero coupon bond is zero”. Unfortunately, the interviewer was
plainly wrong, as the discussion in this section demonstrates.
CONVEXITY
123
$83.18, and deposits the cash amount with the repo dealer. This assumption implies
that the short position is achieved at zero cost for the corporation.4
Consider now a parallel shift in the yield curve. Let’s explore three scenarios: A
small parallel shift with a change of dr = 10 basis point, a medium-large parallel shift
with dr = 1%, and a very large parallel shift with a change of dr = 2%. How does
the duration hedge perform under these three scenarios? To answer this question,
we need to recompute the values of the bonds P and Pz (0, T ) for the new interest
rate scenarios, compute the new value of the portfolio V = P + k × Pz (0, T ), and
then take the difference from the original portfolio value. This exercise is illustrated
in Panel A of Table 4.2. Column 1 reports the size of the change in interest rate.
Columns 2 and 3 report the values of the bond P and the zero coupon bond Pz (0, T ),
respectively, for each interest rate scenario, while Column 4 shows the position in the
zero coupon bond, which does not change across scenarios. Column 5 finally reports
the change in portfolio value dV . As can be seen, the change in portfolio value is
extremely small when the parallel shift is small, dr = 10 basis points. However,
a larger increase in the yield curve still produces a loss, which increases with the
increase in interest rates. In addition, note that the hedged portfolio loses money both
when interest rates increase and when interest rates decrease.
Example 4.3 shows that the hedged portfolio loses money both when interest rates increase
and when interest rates decrease if the size of these shifts is large. This behavior of the
hedged portfolio is due to the fact that the hedged portfolio is not convexity hedged. Indeed,
we can write the change in the hedged porfolio as follows:5 .
d V = d P + k × d Pz
Substituting for both d P and d Pz the approximation in Equation 4.2 we find
dV
1
= −D × P × dr + × P × C × dr2
2
1
+k × −Dz × Pz × dr + × Pz × C × dr2
2
(Change d P )
(Change k × d Pz )
Pulling together the terms in dr and dr2 we obtain
dV
= − (D × P + k × Dz × Pz ) × dr +
1
× (P × C + k × Pz × C) × dr2
2
(4.16)
Duration hedging (Equation 4.15) eliminates the first parenthesis in Equation 4.16. However, the second parenthesis in the equation is generally not zero. If the parenthesis is
negative, then the duration hedging strategy tends to generate a loss irrespective of a positive or negative change in interest rate dr because dr2 > 0. Vice versa, if the parenthesis
is instead positive, then the duration hedging strategy tends to generate a gain.
One strategy to hedge against both small and large variations in interest rates is to choose
both k and the maturity T of the zero coupon bond to make both parentheses in Equation
4 Alternatively,
the corporation can use forward contracts or futures contracts, which also have zero cost, as
discussed in Chapters 5 and 6.
5 For notational simplicity, we denote here the zero coupon bond by P rather than the more cumbersome P (0, T ).
z
z
124
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
Table 4.2 Duration Hedging versus Duration and Convexity Hedging
Panel A: Duration Hedging
Spot Curve
Shift
P
Pz (0, T )
Position
Change in
Portfolio Value
Initial Values
dr = 0.1%
dr = 1.0%
dr = 2.0%
dr = −0.1%
dr = −1.0%
dr = −2.0%
103.58
102.75
95.63
88.38
104.41
112.29
121.84
63.76
63.13
57.69
52.20
64.40
70.47
77.88
-1.3045
-1.3045
-1.3045
-1.3045
-1.3045
-1.3045
-1.3045
-0.0003
-0.0318
-0.1210
-0.0003
-0.0350
-0.1474
Panel B: Duration and Convexity Hedging
Spot Curve
Shift
P
Pz (0, T1 )
Pz (0, T2 )
Postion
k1
Position
k2
Change in
Portfolio Value
Initial Value
dr = 0.1%
dr = 1.0%
dr = 2.0%
dr = −0.1%
dr = −1.0%
dr = −2.0%
103.58
102.75
95.63
88.38
104.41
112.29
121.84
91.39
91.21
89.58
87.81
91.58
93.24
95.12
63.76
63.13
57.69
52.20
64.40
70.47
77.88
-0.4562
-0.4562
-0.4562
-0.4562
-0.4562
-0.4562
-0.4562
-1.1737
-1.1737
-1.1737
-1.1737
-1.1737
-1.1737
-1.1737
0.0000
0.0003
0.0023
0.0000
-0.0003
-0.0027
125
CONVEXITY
4.16 equal to zero. Alternatively, we can use two securities to hedge both duration and
convexity simultaneously. The latter strategy is simpler to implement because we can then
choose freely from all possible available securities, including derivative instruments such
as forwards or futures, discussed in Chapters 5 and 6.
ore specifically, let P1 and P2 be the prices of two securities, such as a short-term and
a long-term zero coupon bond, with D1 , D2 , C1 , and C2 their durations and convexities,
respectively. Let k1 and k2 be the position in these two bonds. The value of the hedged
portfolio is then given by
V = P + k1 × P1 + k2 × P2
The portfolio is hedged if a change in interest rate dr does not affect its value, that is, if
dV = 0. Taking into account also the convexity effect in Equation 4.2, we obtain:
dV
= d P + k1 × d P1 + k2 × d P2
1
= −D × P × dr + × C × P × dr2
2
1
−k1 × D1 × P1 × dr + × k1 × C1 × P1 × dr2
2
1
−k2 × D2 × P2 × dr + × k2 × C2 × P2 × dr2
2
(Change dP in original bond)
(Change dP1 in bond 1)
(Change dP2 in bond 2)
We can pull together the terms in dr and dr2 :
dV
=
− (D × P + k1 × D1 × P1 + k2 × D2 × P2 ) × dr
1
+ × (C × P + k1 × C1 × P1 + k2 × C2 × P2 ) × dr2
2
Thus, in order for the portfolio to be immune to changes in interest rates, we must have
dV = 0 for both small changes in interest rates (small dr) and large changes in interest
rates (large dr which implies large dr2 ). This is achieved by choosing k1 and k2 such that:
k1 × D1 × P1 + k2 × D2 × P2
=
−D × P
(Delta Hedging)
k1 × C1 × P1 + k2 × C2 × P2
=
−C × P
(Convexity Hedging)
The solution of this system of two equations in two unknowns is
D × C2 − C × D2
P
×
k1 = −
P1
D1 × C2 − C1 × D2
D × C1 − C × D1
P
×
k2 = −
P2
D2 × C1 − C2 × D1
(4.17)
(4.18)
We now return to Example 4.3 and see the improvement in hedging performance.
EXAMPLE 4.4
Consider again Example 4.3, but assume that in addition to the zero coupon bond
with maturity T2 = 10, the corporation also uses a zero coupon bond with short
maturity T1 = 2. From Table 4.1 the price of the 2-year zero coupon bond is
Pz (0, T1 ) = $91.39. From the same table, the convexity of the bond we want to
126
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
hedge is C = 73.87. Denoting by D1 = 2, D2 = 10, C1 = 4, and C2 = 100 the
durations and convexities of the two zero coupon bonds, respectively, applying the
formulas in Equations 4.17 and 4.18 we obtain
k1 = −0.4562 and k2 = −1.1737.
That is, to hedge the portfolio against both small and large changes in interest rates,
the corporation must short 0.4562 units of the 2-year zero coupon bond, and 1.1737
units of the 10-year zero coupon bond.
Does the hedging strategy work any better than the simpler duration hedging
strategy? Panel B of Table 4.2 illustrates the performance of the hedging strategy
under the three scenarios of a small, medium-large, and very large shift in the yield
curve. In particular, Column 1 displays the interest rate scenarios, Columns 2 to 4
contain the values of the original bond, the 2-year zero coupon bond, and the 10-year
zero coupon bond, respectively. Columns 5 and 6 show the positions in the short-term
and long-term bond, respectively. Finally, Column 7 contains the change in value
of the hedged portfolio. As we would expect, the changes of the hedged portfolio
are much smaller than under duration hedging in Panel A, even for a relatively large
variation in the term structure of interest rates.
4.1.7
Convexity Trading and the Passage of Time
Example 4.3 and the discussion following it highlight a seemingly profitable trading strategy. If we go long bonds with high convexity, such as long-term bonds, and duration hedge
the interest rate risk using bonds with low convexity, such as short-term bonds, we gain
positive returns from convexity. That is, in Equation 4.16 the first parenthesis is always
zero, because of hedging, while the second parenthesis is always positive, entailing a positive flow of money. For instance, a standard convexity trading strategy is the barbell-bullet
portfolio strategy: A barbell bond position consists of a portfolio that is long both high
duration and low duration assets, while a bullet bond position is a position long a medium
duration asset. The strategy then consists in going long a barbell position, and hedging it
with a bullet position, with the same duration. Such a strategy results in a positive convexity
strategy.
The question is then whether the convexity trading strategy represents an arbitrage
opportunity. The answer is no. The reason is that the duration hedging argument leaves
out an important dimension of bond return, and this is time. That is, a zero coupon bond,
for instance, gains value over time simply because time passes, even if interest rates do not
move. This predictable part in the variation of bond prices is not taken into account in the
duration/convexity hedging strategies discussed above, but it must be taken into account if
we want to consider a dynamic investment strategy. Unfortunately, trading strategies that
also take into account the price variation due to the passage of time require the development
of more complex models, such as those discussed in Part II and Part III of this book. From
these models we discover that the convexity trading strategy has an important drawback,
and this is the fact that the gain in value from higher convexity is offset by a lower gain due
to the passage of time, a relation that is known as the Theta-Gamma relation (see Section
16.5 in Chapter 16). In other words, there is really one more term in Equation 4.16 showing
the changes in value in the bond P and in the zero coupon bond Pz due to time, and these
changes exactly offset the convexity gain.
SLOPE AND CURVATURE
4.2
127
SLOPE AND CURVATURE
In Chapter 3 we looked at the level of interest rates over time, and discussed the fact that
all interest rates tend to go up and down together. That is, the general level of interest rates
is the first quantity to watch out for when we either invest in fixed income securities or
engage in risk management practices. Panel A of Figure 4.6 plots the time series of the
term structure of interest rates in the United States, from 1965 to 2005.
Figure 4.6
Slope and Curvature of the Term Structure of Interest Rates: 1965 - 2005
Panel A. Term Structure of Interest Rates
Percent
20
15
10
5
0
1965
1970
1975
1980
1985
1990
1995
2000
2005
1995
2000
2005
1995
2000
2005
Panel B. Term Spread (Slope)
Percent
5
0
−5
1965
1970
1975
1980
1985
1990
Panel C. Buttefly Spread (Curvature)
Percent
5
0
−5
1965
1970
1975
1980
1985
1990
Data Source: CRPS.
A look at Panel A of Figure 4.6, however, also reveals that the term structure of interest
rates is not moving up and down in a parallel fashion. If it was moving in a parallel
fashion, we should see lines at equal distances over time. Instead, what we see in Panel
A of Figure 4.6 is that the interest rates sometimes are further apart from each other, as
128
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
they were around 1993 and 2003, while other times they are very close to each other, as
in 1989, 1995 and 2005. Indeed, Panel B of the figure plots the term spread of the term
structure of interest rates over time. Recall that the term spread was introduced in Chapter
2, Definition 2.4, and it is given by the difference between a long-term interest rate (in the
figure, the 10-year rate) and a short-term interest rates (in the figure, the 1-month interest
rate). Clearly, the term spread greatly fluctuates over time, moving between -5% to 5%.
This movement may or may not be accompanied by changes in the level of interest rates.
For instance, comparing Panel B of Figure 4.6 with Panel B of Figure 3.1, we see that in
1993 and 1998, the level (average) of interest rates was about 5%. However, in 1993, the
term spread (slope) was around 4% while in 1998, it was around 0%.
A second quantity that is also time varying but not fully accounted for by the level
and the slope of the term structure is the curvature of the term structure of interest rates,
that is, the relative pricing of short-, medium- and long-term bonds. Figure 4.7 provides
an example of two term structures of interest rates, with approximately the same level of
interest rates and the same slope (term premium), but two different curvatures.
Figure 4.7 The Slope and Curvature of the Term Structure at two Dates
6.5
Interest Rate (%)
6
5.5
10/31/1996
08/29/1997
5
0
1
2
3
4
5
6
Time to Maturity
7
8
9
10
Data Source: CRPS.
ore explicitly, the term structure at the end of August 1997 appears more curved than the
one at the end of October 1996. One way to measure the curvature of the term structure of
interest rates is to consider the relative pricing of short-, medium- and long-term bonds. A
popular measure of the curvature of the term structure is called the butterfly spread, defined
as follows:
Definition 4.3 A butterfly spread is given by the following quantity
Curvature = −Short-term yield + 2 × Medium-term yield − Long-term yield
SLOPE AND CURVATURE
Table 4.3
129
The Slope and Curvature of the Term Structure at Two Dates
Date
1-month Yield
5-year Yield
10-year Yield
Slope
Curvature
10/31/1996
08/29/1997
5.01 %
5.05 %
6.08 %
6.21%
6.30%
6.29 %
1.29 %
1.24 %
0.85%
1.08 %
Notes: Yields are calculated based on data from CRSP.
To understand this measure of curvature, consider again Figure 4.7: Table 4.3 reports
the actual yields on the two dates, the slope and curvature measures. We see that while the
slope is almost identical, the curvature computed from the butterfly spread is much higher.
4.2.1
Implications for Risk Management
Why is all this important for risk management? In Chapter 3 we considered the duration
based risk management practice of hedging against parallel shifts of the term structure
of interest rates. That is, hedging only against changes in the average level of interest
rates. However, interest rates do not move in a parallel fashion. This is important: A
bond portfolio that could be duration hedged may still suffer large losses if the slope or the
curvature of the term structure changes. The next example demonstrates the problem.
EXAMPLE 4.5
Suppose that on April 1, 2004 a fixed income fund has $100 million (par amount)
invested in a 3.875 coupon bond expiring on February 15, 2013. The price of this
bond on that day is $101.50 (per $100 of face value). The term structure of interest
rates on that day is displayed in Table 4.4. With these data, the duration of the bond
is equal to
D = 7.491
(4.19)
An effective duration hedge can be achieved by using, for instance, the zero coupon
bond maturing on February 15, 2005. The duration of the zero coupon bond is
DS = 0.87. Duration hedging is then achieved by taking a position kS in the
short-term zero coupon bond equal to
kS = −
D×P
7.491 × 101.5
= −8.83
=
DS × PS
0.87 × 99.0019
That is, the fund must short 8.83 short-term bonds for each unit of long-term bond
held. Assume that the short position kS in short-term bonds is taken through the repo
market, so that any cash inflow is deposited with the repo dealer and the net position
of the fund is simply equal to V = $101.5.
What happened between April 1, 2004 and April 15, 2004? The term structure
of interest rates indeed shifted up by about 0.5%, on average, across all maturities.
130
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
The two term structures on the two dates are plotted in Figure 4.8, and also reported
in Columns 3 and 6 of Table 4.4, respectively. An effective duration hedging should
have largely hedged the potential losses due to the shift in interest rates. This can be
seen by direct computation: In fact, we can add 0.50% to each of the interest rates
in the second column of Table 4.4 and recompute the value of the hedged portfolio.
We find that such a parallel shift in the term structure of interest rates would change
the value of the hedged portfolio from $101.50 to $101.57, implying a small (0.07%)
increase in value.
In sharp contrast, the new term structure of interest rates implies a large negative
drop in price of the portfolio. In fact, the value of the hedged portfolio using the new
term structure interest rates contained in Column 5 of Table 4.4 is
V4/15/2004 = $98.20,
a decline of 3.30%. In other words, the hedging strategy did not work as expected.
It did work partially, as the unhedged portfolio would have dropped instead to
no hedge
V4/15/2004 = $97.42,
a decrease of 4.01%. But the fact that the allegedly hedged portfolio also dropped
substantially in value is quite bad for the duration hedging strategy.
The previous example highlights that the hedging of interest rate risk cannot be fully
accomplished by using the notion of duration. More specifically, from Figure 4.8, the
reason why the duration hedged portfolio in Example 4.5 dropped in value is that between
April 1 and April 15 the slope of the term structure changed as well. Since the long end
increased by more than the short end, the price of the security used to hedge (i.e., the
short-term bond) did not increase sufficiently compared to the drop of the long-term bond.
Therefore, we must extend our risk management methodology to take into account changes
in the slope and curvature of the term structure.
4.2.2 Factor Models and Factor Neutrality
How can we expand the duration methodology to take into account that slope and curvature
can change over time? Let’s begin by considering level, slope and curvature as factors that
drive the term structure of interest rates.
Definition 4.4 Consider T1 , T2 ,...,Tn to be n points on the current term structure of interest
rates, and let ri = r(t, Ti ) be the corresponding zero coupon rate. A factor model for the
dynamics of the term structure of interest rates assumes that the instantaneous change of
the various points on the curve, dri , is due to a set of common factors φ1 , φ2 ,...,φm
dr1
=
β 11 dφ1 + β 12 dφ2 + ...β 1m dφm
(4.20)
dr2
..
.
=
..
.
β 21 dφ1
..
.
+ β 22 dφ2 + ...β 2m dφm
..
.
(4.21)
drn
=
β n 1 dφ1 + β n 2 dφ2 + ...β n m dφm
(4.22)
where β ij determines the impact that the variation in each of the factors dφj , j = 1, .., m
has on each individual interest rate ri , i = 1, ..., n.
SLOPE AND CURVATURE
Table 4.4
aturity
Date T
Time to
Maturity T
20040815
20050215
20050815
20060215
20060815
20070215
20070815
20080215
20080815
20090215
20090815
20100215
20100815
20110215
20110815
20120215
20120815
20130215
0.37
0.87
1.37
1.87
2.37
2.87
3.37
3.87
4.37
4.87
5.37
5.87
6.37
6.87
7.37
7.87
8.37
8.83
Interest Rates in April, 2004
1 April 2004
Yield
Discount
r(0,T) Z(0, T )(×100)
1.00%
1.15%
1.32%
1.55%
1.79%
2.01%
2.22%
2.44%
2.66%
2.83%
3.02%
3.19%
3.32%
3.44%
3.59%
3.69%
3.77%
3.86%
131
15 April 2004
Time to
Yield
Discount
Maturity T − t r(t, T ) Z(0, T )(×100)
99.6293
99.0019
98.2068
97.1458
95.8461
94.3856
92.7813
90.9886
89.0279
87.1094
85.0175
82.9229
80.9208
78.9194
76.7659
74.8008
72.9193
71.1246
0.33
0.83
1.33
1.83
2.33
2.83
3.33
3.83
4.33
4.83
5.33
5.83
6.33
6.83
7.33
7.83
8.33
8.79
1.12%
1.31%
1.67%
1.98%
2.28%
2.55%
2.77%
2.99%
3.22%
3.40%
3.62%
3.79%
3.90%
4.03%
4.16%
4.25%
4.33%
4.43%
99.6283
98.9140
97.7961
96.4300
94.8167
93.0378
91.1796
89.1559
86.9737
84.8449
82.4343
80.1709
78.0958
75.9416
73.6930
71.6587
69.6940
67.7355
Notes: Yields and discounts are calculated based on data from CRSP.
Figure 4.8 The Shift Up in the Term Structure in April, 2004
4.5
4
3.5
Yield (%)
3
2.5
2
1.5
1 April 2004
15 April 2004
1
0.5
0
0
1
Data Source: CRSP.
2
3
4
5
Time to Maturity
6
7
8
9
132
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
Table 4.5 The Sensitivity of Interest Rates to Level, Slope, and Curvature
Maturity
2 year
4 year
Factor
3 month
6 month
1 year
β i 1 (Level)
β i 2 (Slope)
β i 3 (Curv.)
1.0967
-0.1794
-0.3184
1.1686
-0.1506
-0.1131
1.1154
-0.2540
0.1761
0.9940
-0.3010
0.3723
R2
97.56%
92.45 %
96.16%
99.99%
6 year
8 year
10 year
0.9333
0.0080
0.1790
0.9264
0.3158
-0.1076
0.9267
0.5297
-0.3224
0.9279
0.6737
-0.4708
97.98%
96.90%
96.57%
96.56%
To clarify Definition 4.4, in the case of duration-based risk management we considered
only parallel shifts. That is, we considered only one factor (m = 1 above), called parallel
shift, and all of the β i1 = 1. In this case, an increase in the “factor φ1 ” generates an
increase in the overall term structure of interest rates at all points T1 ,...,Tn . Definition
4.4 generalizes this case to multiple factors, in which φ2 could be slope and φ3 could be
curvature, as plotted in Figure 4.6.6
How do we use a model such as that supplied by Equations 4.20 through 4.22? Suppose
that we have available the β ij coefficients. Then, we can neutralize the impact of changes
in factors, such as level, slope and curvature, much in the same way as we did for duration.
To be concrete, consider the case in which φ1 = level, φ2 = slope and φ3 = curvature.
Table 4.5 contains estimates of the sensitivities β ij for various maturities T1 = 3-months,
T2 = 6-months, ... , Tn = 10-years. We will discuss later how to obtain these coefficients.
For now, assume we have them. The question is how do we use this information to quantify
risk and set up an effective risk management strategy.
4.2.3 Factor Duration
uch in the same way as in Chapter 3 when we computed the sensitivity of bond prices to
parallel shifts in the yield curve (duration analysis), we can compute the sensitivity of a
bond price to each of the factors (level, slope, and curvature). The methodology is very
similar. Consider first the notion of factor duration, which is analogous to parallel shifts
but defined on the factors themselves:
Definition 4.5 The factor duration of an asset with price P with respect to factor j (e.g.,
slope) is defined as
1 dP
(4.23)
Dj = −
P d φj
where
6 An
d P
d φj
represents the sensitivity of the asset price to changes in factor j.
important caveat is the following: As it is true in the notion of duration, there is no time dimension in the
factor model in Equations 4.20 to 4.22. The interpretation of the model is to gauge the impact that factors φ j ,
j = 1, .., m would have on the value of a portfolio if suddenly they moved up or down. This notion is important
to understanding the connection with no arbitrage models described in Part III of the book.
SLOPE AND CURVATURE
133
The factor duration measures the (negative of the) percentage impact of factor j on the
price P . For instance, if the factor is the level, the factor duration is essentially equivalent
to the conventional duration we talked about in Chapter 3. Note that the definition can also
be given also for slope and curvature.7
As we did for the case of duration, we begin by analyzing a zero coupon bond with price
Pz (t, Ti ) and time to maturity (Ti − t). In particular, if factor j (e.g. slope) moves from
φj to φj = φj + dφj , what is the impact on the price of the zero coupon Pz (t, Ti )? The
methodology is as follows: The variation in factor j, dφj , has an impact on the yield ri
that corresponds to the maturity of the zero coupon bond we are analyzing. The size of this
impact of φj on ri is given by the coefficient β ij in Equations 4.20 - 4.22, and provided
in Table 4.5. That is, assuming that only factor j moves, dri = β ij dφj . Since dri has an
impact on the zero coupon bond price given by its first derivative with respect to the yield
to maturity, we can compute the total sensitivity of the bond price to changes in the factor.
More specifically, this chain rule, or domino effect (the factor has an impact on interest rate
that has an impact on price) can be expressed mathematically as follows:
dPz (t, Ti )
dri
dPz (t, Ti )
dPz (t, Ti )
=
×
=
× β ij
dφj
dri
dφj
dri
(4.24)
i)
= −(Ti − t) × Pz (t, Ti ) we obtain the factor duration of the zero
Recalling that dP zdr(t,T
i
coupon i with respect to factor j:
Dj,z
dPz (t, Ti )
1
Pz (t, Ti )
dφj
1
−(Ti − t) × Pz (t, Ti ) × β ij
= −
Pz (t, Ti )
= (Ti − t) × β ij
=
−
(4.25)
The formula is the same as for the duration, with the additional twist that β ij enters
into the picture. Note that given the parameters in Table 4.5 we have β i1 ≈ 1 for the level
factor, thereby obtaining the standard definition of duration for the first factor.
Since the equation for the duration of a zero coupon bond is similar to the one for the
standard duration (with the addition of β ij ) it is intuitive that all of the other results we
obtained in Chapter 3 also hold. In particular:
Fact 4.4 The factor duration of a portfolio is the weighted average of the factor duration
of the individual assets, where the weights equal the relative value of each asset with respect
to the total value of the portfolio.
For instance, as an application of this fact, we obtain:
Fact 4.5 Consider a coupon bond with price Pc (t, T ), maturity T , coupon c, and payment
dates T1 , T2 ,...,Tn . The factor duration of a coupon bond with respect to factor j is given
by
n
Dj =
wi × (Ti − t) × β ij
(4.26)
i= 1
7 The
negative sign in front of the expression in Equation 4.23is for symmetry with respect to the definition of
duration in Chapter 3.
134
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
where wi = c/2 × Pz (t, Ti )/Pc (t, T ) for i = 1, .., n − 1 and wn = (1 + c/2) ×
Pz (t, Tn )/Pc (t, T )
The notion of factor duration allows us to compute the percentage variation of bond
prices due not only to changes in level of interest rates (as in standard duration), but also
due to changes in the slope and curvature. In particular, we have the following fact:
Fact 4.6 Consider an asset with price P and factor durations D1 , D2 , and D3 with respect
to the three factors φ1 , φ2 , and φ3 . Then, the percentage change in price is approximately
equal to
dP
= −D1 × dφ1 − D2 × dφ2 − D3 × dφ3
(4.27)
P
EXAMPLE 4.6
As in Example 4.5, on April 1, 2004, consider the 3.875% bond maturing on February
15, 2013. Given the term structure of interest rates on that date in Table 4.4, the data
in Table 4.5, Equation 4.26 yields the following factor durations:
D1 = 6.9624; D2 = 4.0797; D3 = −2.5741
(4.28)
These numbers can be interpreted as follows: First, D1 has the same meaning as
duration (and in fact, the value is very similar to the one obtained in Equation
4.19), namely, an increase in the average level of interest rates by one basis point,
dφ1 = 0.01%, will decrease the bond price by 0.069624%. Similarly, D2 implies
that an increase in slope by dφ2 = 0.01% will decrease the bond price by 0.040797%.
Finally, D3 implies that an increase in curvature by dφ3 = 0.01% will increase the
bond price by 0.025741%.8
4.2.4
Factor Neutrality
Consider now a portfolio P with factor durations D1 and D2 with respect to level and slope
factors φ1 and φ2 , respectively.9 To implement factor neutrality, we need to select two
other securities, one for each factor we want to neutralize, in appropriate proportions. For
instance, we could use short- and a long-dated zero coupon bonds, denoted by PzS and PzL .
For each of these two bonds we can compute the factor durations. Let’s denote by DzS,1 and
DzS,2 the factor durations of the short-dated zero coupon bond, and by DzL,1 and DzL,2 those
of the long-dated zero coupon bond. In order to immunize the portfolio against changes in
level and slope, we must choose a number of short-term and long-term zero coupon bonds,
kS and kL , such that the variation of the portfolio plus the two bonds is approximately zero.
8 One
drawback of defining level, slope and curvature as average interest rate, term spread, and butterfly spread,
respectively, is that changes in level, slope and curvature are not independent from each other. For instance, when
the term structure is strongly upward sloping, a decline in slope is also accompanied by a decline in curvature, as
the term structure flattens out. The appendix at the end of this chapter provides a more advanced methodology to
obtain independent factors with the same interpretation.
9 For convenience, we only illustrate the case with two factors. The methodology with three factors is identical.
SLOPE AND CURVATURE
135
That is, such that the change in V = P + kS × PzS + kL × PzL satisfies
d V = d P + kF × d PzS + kL × d PzL = 0
We can substitute into d P , d PzS and d PzL the equivalent of expressions in Equations 4.27
with only two factors, obtaining the following equation:
0
= −D1 × P × dφ1 − D2 × P × dφ2
+kS × (−DzS1 × PzS × dφ1 − DzS2 × PzS × dφ2 )
+kL × (−DzL1 × PzL × dφ1 − DzL2 × PzL × dφ2 )
Pool together all the elements containing dφ1 and dφ2
0
= −(D1 × P + kS × DzS1 × PzS + kL × DzL1 × PzL ) × dφ1
−(D2 × P + kS × DzS2 × PzS + kL × DzL2 × PzL ) × dφ2
In order for the equation to be zero for all possible values of dφ1 and dφ2 , each parenthesis
on the right-hand side must be zero. We therefore obtain a system of two equations in two
unknowns:
kS × DzS1 × PzS + kL × DzL1 × PzL = −D1 × P
kS × DzS2 × PzS + kL × DzL2 × PzL = −D2 × P
The solution of this system is:
kS
kL
=
=
P
− S
Pz
−
P
PzL
D1 × DzL2 − D2 × DzL1
DzS1 × DzL2 − DzS2 × DzL1
D1 × DzS2 − D2 × DzS1
DzL1 × DzS2 − DzL2 × DzS1
(4.29)
(4.30)
Example 4.5 shows the shortcomings of duration hedging. We now follow up with that
example to highlight the better hedging performance achieved by factor neutrality.
EXAMPLE 4.7
As discussed, the reason of the underperformance of the duration hedged portfolio in
Example 4.5 is the change in shape of the term structure between April 1 and April
15, 2004. In particular, the shift was not parallel, as long-term yields shifted up by
more than short-term yields. Since hedging was performed using a short-term bond,
the change in the shape of the term structure generated the losses.
Hedging against changes in slope mitigates the problem. In particular, we now
apply the hedging strategy in Equations 4.29 and 4.30 using the zero coupon bonds
maturing on February 15, 2005 and on February 15, 2013. Because we are using
zero coupon bonds, we can directly apply Equation 4.25 for the computation of factor
durations. Given the data in Tables 4.4 and 4.5, we obtain
D1S = 0.9729; D2S = −0.2215
D1L
= 8.1912;
D2L
= 5.3150
(4.31)
(4.32)
136
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
Using Equations 4.29 and 4.30 we obtain that the positions kS and kL in the short-term
and long-term zero coupon bonds are
kS = −0.5266; kL = −1.1259
As in Example 4.5, assume that the short positions in these short- and long-term
zero coupon bonds occur through the repo market. The question is then whether this
hedged portfolio was better able to weather the increase in the interest rates in the
first two weeks of April 2004. The answer is yes. Using the data on the right-hand
side of Table 4.4, we obtain a value of the hedged portfolio
H edg ed
= 100.9076,
V4/15/2004
implying a drop in value of only 0.58%. We must compare this number with the drop
in value of the duration hedged portfolio, which was about 3.30%.
4.2.5
Estimation of the Factor Model
Where do the β ij in Table 4.5 come from? They can be estimated by using historical data
on yields and the factors. In particular, we can proceed as follows: Let τ 1 , τ 2 ,..., τ n be
n given times to maturity, which we keep fixed. For instance, τ 1 = one month, τ 2 = 2
months, ... ,τ n = 10 years. We are interested in finding out how much a change in each of
the factors φj affects the rates at these maturities. Let h be the time interval corresponding
to our historical data. For instance, if we have daily data, h = 1/252, while if we have
monthly data, then h = 1/12. For each time to maturity τ i , consider the change in the zero
coupon bond yield r(t, t + τ i ) between t and t + h,
Δri (t) = r(t + h, t + h + τ i ) − r(t, t + τ i )
So, if τ i = 1 year, then Δri (t) is the change of the 1-year yield during the time interval h.
Similarly, denote by φj (t) factor j. As in the previous section, we think of φ1 (t) = level,
φ2 (t) = slope and φ3 (t) = curvature, whose time series are plotted in Panel B of Figure
3.1 in Chapter 3 and Panels B and C in Figure 4.6. Using these data, we approximate the
factor model (Equations 4.20 through 4.22) as follows:
Δr1 (t) = α1 + β 11 Δφ1 (t) + β 12 Δφ2 (t) + β 13 Δφ3 (t) + ε1 (t)
Δr2 (t) = α2 + β 21 Δφ1 (t) + β 22 Δφ2 (t) + β 23 Δφ3 (t) + ε2 (t)
..
.
. = ..
(4.33)
Δrn (t) = αn + β n 1 Δφ1 (t) + β n 2 Δφ2 (t) + β n 3 Δφ3 (t) + εn (t)
where ε1 (t),... ,εn (t), are random errors denoting the fact that our factor model is unlikely
to be a perfect model of the variation of interest rates, and α1 ,...,αn represent the average
change in interest rates that is not due to the assumed factors (these numbers are small).
Given data on yields ri (t) and factors φj (t), we can estimate the coefficient β ij by
using a regression analysis. As we might anticipate, Table 4.5 contains the results of this
SUMMARY
137
regression analysis. It is worth mentioning a few interesting features of the coefficients
β ij : The first row contains β i1 , the sensitivity of yields to the first factor, the level factor.
Since all of the yields move up and down in tandem, we would expect the coefficient to be
relatively constant across maturities, and close to one. And indeed, this is the case.
The second row contains the estimates of β i2 , the sensitivities of yields to the second
factor, the slope factor. The coefficients are negative for low maturities and positive for
high maturities. The interpretation is that if this factor kicks in it lowers the short-term
yields and increases the long-term yields. That is, the slope increases. Finally, the third
row contains the estimates of β i3 , the sensitivities of yields to the curvature factors. The
coefficients are negative for short- and long-term bonds, and positive for medium-term
bonds. Again, when the curvature factor increases it decreases the short- and long-yields,
but it increases the medium-term ones. In other words, this factor bends the term structure
of interest rates.
The key insight for risk management from this exercise is in fact contained in the fourth
row of Table 4.5, which reports the R2 from the regression. The R2 quantifies the power of
the three factors to explain the movements of the various points of the term structure over
time. For instance, R2 = 99.99% for the 4-year interest rate means that the changes in the
4-year yield are essentially entirely explained by the three factors level, slope and curvature
discussed earlier. A glance to the R2 ’s shows that these three factors explain most of the
variations of all of the yields. This fact has an important implication for risk management:
An effective risk management strategy can be attained by using only three securities in
proportions chosen to hedge the three factors. For instance, a financial institution that
holds bonds with many different maturities can achieve an effective risk management of its
assets’ variation in prices by using only three other securities, such as zero coupon bonds
or interest rate derivatives.
4.3
SUMMARY
In this chapter we covered the following topics:
1. Convexity: The rate of decrease in bond prices due to a parallel shift in the term
structure declines as we increase interest rates is called converxity. That is, if the level
of interest rates rises, bond prices decline by less than their predicted decline from
duration. Vice versa, if the level of interest rates declines, bond prices increase by
more than their predicted increase from duration. Convexity measures this difference.
The convexity of a zero coupon bond is its time to maturity squared.
2. Convexity of a portfolio: As with duration, the convexity of a portfolio can be
computed as the weighted average of the convexities of the securities in the portfolio,
where the weights are the percentage position of the securities in the portfolio.
3. Term Structure Slope: The term structure of interest rates not only changes level
over time, but also shape. In particular, the spread between long-term and short-term
yields, the slope of the term structure, changes over time.
4. Term Structure Curvature: The term structure has curvature whenever the short-,
medium-, and long-term rates do not lie on a straight line. A hump-shaped term
structure has positive curvature. The curvature of the term structure, the relative
138
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
Table 4.6
Yield Curve on May 15, 2000
aturity
Yield
Maturity
Yield
Maturity
Yield
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
6.33%
6.49%
6.62%
6.71%
6.79%
6.84%
6.87%
6.88%
6.89%
6.88%
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
6.86%
6.83%
6.80%
6.76%
6.72%
6.67%
6.62%
6.57%
6.51%
6.45%
5.25
5.50
5.75
6.00
6.25
6.50
6.75
7.00
7.25
7.50
6.39%
6.31%
6.24%
6.15%
6.05%
5.94%
5.81%
5.67%
5.50%
5.31%
Notes: Yields are calculated based on data from CRSP.
yield of medium coupon bonds with respect to the long term and the short term,
changes over time.
5. Factor duration: This is the (negative of the) sensitivity of a security price to changes
in a factor (level, slope and curvature), in percentage.
6. Factor neutrality: Duration hedging is only with respect to the level of interest rates.
Factor neutrality is the active hedging of slope and curvature in addition to the level
of interest rates. It helps in particular for portfolios that are tilted toward the short
term or the very long term.
7. Principal Component Analysis: This advanced methodology, discussed in the appendix to the chapter, extracts independent factors (level, slope, and curvature) from
the term structure of interest rates. It helps to obtain more effective hedge ratios.
4.4 EXERCISES
1. On May 15, 2000 the semi-annually compounded yield curve was as in Table 4.6.
Calculate the convexity for the following securities:
(a) 4-year zero coupon bond
(b) 2 1/4-year coupon bond paying 5% semiannually
(c) 2-year coupon bond paying 3% quarterly
(d) 3 1/2-year floating rate bond with 20 basis point spread, paid semiannually
(e) 4 1/4-year floating rate bond with 35 basis point spread, paid semiannually
2. It is May 15, 2000 and an investor is planning to invest $100 million in one of the
two portfolios below. The investor’s main concern is the change in interest rates that
might affect the short-term value of the portfolio. Compute the change in price of
the security stemming from duration and convexity. Which portfolio is less sensitive
to changes in interest rates? The portfolios are the following:
EXERCISES
139
Portfolio A
•
•
•
•
30% invested in 5-year coupon bonds paying 4% quarterly
25% invested in 4 1/4-year coupon bonds paying 6% semiannually
20% invested in 90-day zero coupon bonds
15% invested in 2 1/2-year floating rate bonds with zero spread paid
quarterly
• 10% invested in 6-year zero coupon bonds
Portfolio B
• 40% invested in 7-year coupon bonds paying 2% semiannually
• 30% invested in 3 1/4-year floating rate bonds with 50 basis point spread
paid semiannually
• 20% invested in 4-year coupon bonds paying 3.5% semiannually
• 10% invested in 90-day zero coupon bonds
3. Consider Exercise 2. You are told that the mean of daily change in interest rates is
zero and that the variance of the daily change of interest rates is 3.451 × 10 − 7.
What is the annualized expected return taking into account convexity?
4. Rework Example 4.3 but using a 2-year zero coupon bond for hedging, instead of
the 10-year zero coupon bond. How do the results in Table 4.2 change?
5. Consider the trade of purchasing a 10-year coupon bond and hedge the interest rate
risk using a 2-year zero coupon bond. Assume the term structure of interest rates is
flat at the 4.5% continuously compounded interest rate. Compute the profits/losses
from the strategy under various scenarios of interest rate variation, such as a positive
or negative shift of 10 basis points, 1%, or 2% (see e.g., Example 4.3). Perform
this exercise assuming (a) The trade is performed over one day; (b) The trade is
performed over one week; (c) The trade is performed over one month. How do the
results change under these various scenarios? Discuss your results.
6. Compute the level factor and the butterfly spread for each term structure in Table 4.7.
(a) Which period had the highest slope?
(b) Which period had the highest curvature?
(c) Can you recognize the periods with higher slope or curvature on a graph?
(d) Which interval saw the greatest change in slope?
(e) Which interval saw the greatest change in curvature?
7. Using Tables 4.8 and 4.9, compute the factor duration of level, slope, and curvature,
for each of the following securities on February 15, 1994:
(a) 4-year zero coupon bond
(b) 2 1/2-year coupon bond paying 3% semiannually
(c) 3 1/4-year floating rate bond with zero spread paid semiannually
(d) 4 1/4-year floating rate bond with 35 basis point spread paid semiannually
140
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
Table 4.7 Term Structures
DATE
1 month
3 month
6 month
1 year
2 year
3 year
5 year
7 year
10 year
9/26/2008
9/10/2008
8/25/2008
8/11/2008
7/25/2008
7/10/2008
6/25/2008
6/10/2008
0.21%
1.58%
1.66%
1.77%
1.72%
1.48%
1.49%
2.00%
0.87%
1.65%
1.74%
1.87%
1.75%
1.67%
1.81%
2.02%
1.54%
1.87%
1.96%
2.05%
1.95%
2.01%
2.22%
2.24%
1.81%
2.06%
2.12%
2.27%
2.35%
2.20%
2.48%
2.53%
2.11%
2.22%
2.33%
2.56%
2.70%
2.44%
2.82%
2.91%
2.38%
2.42%
2.62%
2.84%
3.01%
2.72%
3.11%
3.20%
3.05%
2.91%
3.04%
3.27%
3.45%
3.10%
3.54%
3.54%
3.41%
3.23%
3.36%
3.57%
3.73%
3.40%
3.78%
3.77%
3.85%
3.65%
3.79%
3.99%
4.13%
3.83%
4.12%
4.11%
Notes: Yields are calculated based on data from CRSP.
Table 4.8
β (Level)
β (Slope)
β (Curvature)
R2
Level, Slope and Curvature
3 month
6 month
1 year
2 year
3 year
5 year
7 year
10 year
1.0180
-0.2568
-0.3284
99.65%
0.9509
-0.3252
-0.1404
99.69%
0.9196
-0.4317
0.0847
98.88%
1.0344
-0.3507
0.3228
99.61%
1.0299
-0.1424
0.3240
99.77%
1.0180
0.2432
0.1716
99.90%
1.0111
0.5205
-0.1058
99.73%
1.0180
0.7432
-0.3284
99.90%
8. In this exercise you need to describe an immunization strategy for a portfolio, given
the factors in Table 4.8. The term structures of interest rates at two dates are in Table
4.9.
(a) You are standing at February 15, 1994 (see table) and you hold the following
portfolio:
• Long $30 million of a 6-year inverse floater with coupon paid quarterly
• Long $30 million of a 4-year floating rate bond with a 45 basis point spread
paid semiannually
• Short $20 million of a 3-year coupon bond paying 4% semiannually
i. What is the total value of the portfolio?
ii. Compute the dollar duration of the portfolio.
(b) You are worried about interest rate volatility. You decided to hedge your
portfolio with the following bonds:
• A 3-month zero coupon bond
• A 6-year zero coupon bond
i. How much should you go short/long on these bonds in order to make the
portfolio immune to interest rate changes?
ii. What is the total value of the portfolio now?
(c) Assume that it is now May 13, 1994 and that the yield curve has changed
accordingly.
EXERCISES
Table 4.9
141
Two Term Structures of Interest Rates
aturity
02/15/94
Yield (c.c.)
02/15/94
Z(t, T )
05/13/94
Yield (c.c.)
05/13/94
Z(t, T )
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
5.25
5.50
5.75
6.00
3.53%
3.56%
3.77%
3.82%
3.97%
4.14%
4.23%
4.43%
4.53%
4.57%
4.71%
4.76%
4.89%
4.98%
5.07%
5.13%
5.18%
5.26%
5.31%
5.38%
5.42%
5.43%
5.49%
5.53%
0.9912
0.9824
0.9721
0.9625
0.9516
0.9398
0.9287
0.9151
0.9031
0.8921
0.8786
0.8670
0.8531
0.8400
0.8268
0.8145
0.8023
0.7893
0.7770
0.7641
0.7525
0.7418
0.7293
0.7176
4.13%
4.74%
5.07%
5.19%
5.49%
5.64%
5.89%
6.04%
6.13%
6.23%
6.31%
6.39%
6.42%
6.52%
6.61%
6.66%
6.71%
6.73%
6.77%
6.83%
6.86%
6.89%
6.93%
6.88%
0.9897
0.9766
0.9627
0.9495
0.9337
0.9189
0.9020
0.8862
0.8712
0.8558
0.8406
0.8255
0.8117
0.7959
0.7805
0.7663
0.7519
0.7387
0.7251
0.7106
0.6977
0.6846
0.6713
0.6619
Notes: Yields and discounts are calculated based on data from CRSP.
142
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
Table 4.10 Factor Sensitivity (1952 - 1993)
Maturity
3m
1y
2y
3y
4y
5y
Parallel: β i , 1
R2
0.4617
0.76
0.4893
0.94
0.4215
0.95
0.3780
0.92
0.3507
0.86
0.3222
0.84
Slope: β i , 2
R2
-0.7868
0.99
-0.1080
0.95
0.1581
0.96
0.2655
0.97
0.3787
0.96
0.3610
0.95
Curvature: β i , 3
R2
0.4047
0.99
-0.7976
0.99
-0.1040
0.96
0.1481
0.97
0.3483
0.97
0.2144
0.96
i. What is the value of the unhedged portfolio now?
ii. What is the value of the hedged portfolio?
iii. Is the value the same? Did the immunization strategy work? How do you
know that changes in value are not a product of coupon payments made
over the period?
(d) Instead of assuming that the change took place 3 months later, assume that the
change in the yield curve occurred an instant after February 15, 1994.
i. What is the value of the unhedged portfolio?
ii. What is the value of the hedged portfolio?
4.5 CASE STUDY: FACTOR STRUCTURE IN ORANGE COUNTY’S
PORTFOLIO
In Chapter 3 we performed a duration analysis of inverse floaters and the portfolio of
Orange County. In this section, we repeat the exercise, but this time to gauge the sensitivity
of the portfolio to the level, slope, and curvature factors described in previous sections. We
will conclude with an application of the factor model to compute the Value-at-Risk of the
portfolio.
4.5.1 Factor Estimation
We use principal component analysis, explained in the Appendix at the end of this chapter,
to compute the sensitivities of interest rates to three factors, using data from 1952 to 1994.
The results are contained in Table 4.10.
4.5.2 Factor Duration of the Orange County Portfolio
We now assume that the Orange County portfolio has $2.8 billion invested in inverse floaters
(see Section 3.8 of Chapter 3), and the remaining part in 3-year zero coupon bonds, for
CASE STUDY: FACTOR STRUCTURE IN ORANGE COUNTY’S PORTFOLIO
Table 4.11
143
Factor Durations of a 3-year Zero Coupon Bond
Factor j
Maturity Ti = 3
βij
Dz , j (3)
1
2
3
3
3
3
0.3780
0.2655
0.1481
1.1340
0.7965
0.4443
simplicity.10 We must compute the factor durations of the portfolio, that is, the sensitivity of
the portfolio to each of the factors. Let DP ,1 , DP ,2 , and DP ,3 be the three factor durations
of the portfolio to level (factor φ1 ), slope (factor φ2 ) and curvature (factor φ3 ). We will
use them to compute the variation in the portfolio changes dP , according to Equation 4.27,
that is
(4.34)
dP = P × (−DP ,1 × dφ1 − DP ,2 × dφ2 − DP ,3 × dφ3 )
We first need to compute the factor duration of the zero coupon bond and the inverse
floater. Table 4.11 shows the computations of the factor durations for the 3-year zero
coupon bond. From Table 4.10 we select the column corresponding to the 3-year maturity
(Column 4) and multiply the corresponding sensitivities β ij by the maturity of the zero,
i.e., Ti = 3.
We now turn to the factor duration of the inverse floater. In this case, we need to use the
same methodology as for the duration of a portfolio. Recall that the inverse floater can be
replicated by a long position in a 3-year fixed rate coupon and a 3-year zero coupon, and a
short position in floating rate bond. The factor duration satisfies the same formula used for
the standard duration (see Equation 3.50 in Chapter 3):
DI F ,j = wz er o × Dz ,j (3) + wf ixed × Dc,j + wf loatin g × DF l,j
where wz er o = 0.7521, wf ixed = 1.1079, and wf loatin g = −0.8600 are the weights
computed in Chapter 3 for the same exercise, and Dc,j and DF l,j denote the factor durations
of the fixed and floating rate bond, respectively. We already have the factor durations
Dz ,j (3) for the 3-year zero coupon bond in Table 4.11. We now need to compute the factor
durations of the fixed rate bond and the floating rate bond. Recalling that a fixed rate bond
is simply a portfolio of zero coupon bonds, Table 4.12 computes the factor durations for
the fixed rate bond.
Finally, we need the factor duration of the floating rate bond. However, as with the
standard duration, this coincides with the factor duration of a zero coupon bond with
maturity equal to the first reset date. In this example, the coupon payments are annual,
and thus the factor duration of the floating rate bond coincides with the factor duration of
a 1-year bond, which is contained in the second column in Table 4.12. Table 4.13 finally
obtains the factor durations of the 3-year inverse floater. It is interesting to compare the
factor durations of the inverse floater to those of a 3-year coupon bond. We see that while
the 3-year coupon bond is strongly affected by parallel shifts (the first factor) but only
10 This
approximation slightly increases the duration of the Orange County portfolio compared to Section 3.8 of
Chapter 3, but it simplifies our calculations, as we only have factor sensitivities at maturity T = 2 and T = 3.
To compute the sensitivity at other maturities we would need an additional interpolation step.
144
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
Table 4.12 Factor Durations of a 3-year Zero Coupon Bond
Factor j
Dz , j (1)
Dz , j (2)
Dz , j (3)
=⇒
Dc , j
1
2
3
0.4893
-0.1081
-0.7976
0.8430
0.3162
-0.2081
1.1340
0.7965
0.4444
=⇒
=⇒
=⇒
1.0305
0.6435
0.2352
0.1123
Weights
0.1070
0.7807
Table 4.13
Factor Durations of Inverse Floater
Factor j
Dz , j (3)
Dc , j
DF l , j
=⇒
DI F , j
1
2
3
1.1340
0.7965
0.4444
1.0305
0.6435
0.2352
0.4893
-0.1081
-0.7976
=⇒
=⇒
=⇒
1.5738
1.4049
1.2807
0.7521
Weights
1.1079
-0.8600
mildly affected by curvature (the third factor), the inverse floater shows a strong sensitivity
across all factors. This means that any change in any of the factors has a large impact on
the inverse floater, compared to the fixed rate bond of the same maturity.
Finally, we can compute the factor duration of the portfolio itself, obtaining
DP ,j
⎧
⎨ 1.1941
2.8
20.5 − 2.8
0.8796
× DI F ,j +
× Dz ,j (3) =
=
⎩ 0.5587
20.5
20.5
(4.35)
4.5.3 The Value-at-Risk of the Orange County Portfolio with Multiple
Factors
We can now apply the same methodology as in Section 3.8 of Chapter 3 and compute the
Value-at-Risk when we decompose the variation in interest rates in multiple factors. For
simplicity, we use only the normal distribution approach. That is, we assume that the three
factors dφ1 , dφ2 , and dφ3 are distributed according to a jointly normal distribution. The
construction of these factors from principal component analysis implies that they are also
independent (see the appendix at the end of the chapter). From Equation 4.34, then, dP
also has a normal distribution, with mean and standard deviation given by
Mean(dP ) = μP
=
P × [−DP ,1 × mean(dφ1 ) − DP ,2 × mean(dφ2 )
−DP ,3 × mean(dφ3 )]
Std(dP ) = σ P
=
P × [DP ,1 × std(dφ1 ) + DP ,2 × std(dφ2 ) + DP ,3 × std(dφ3 )]
APPENDIX: PRINCIPAL COMPONENT ANALYSIS
Table 4.14
mean
st.dev
145
Statistical Properties of the Level of Interest Rate and the Three Factors
dr
dφ1
dφ2
dφ3
4.71E − 05
0.00432
1.0986e-004
0.0107
0
0.0034
0
0.0015
From the time series of factors, we can compute the mean and standard deviations of the
factors dφj , which are in Table 4.14. Using this information, we obtain μP = −0.00269
and σ P = 0.34031. Thus, the 99% monthly VaR is
99% monthly VaR = −μP + 1.634 × σ P = $794 million
Using only the standard duration as in Section 3.8 of Chapter 3 we obtain instead VaR
= $660 million, which is smaller.11 The reason why using only the traditional duration we
obtain a smaller VaR is that in this calculation we use the volatility of the average level
of interest rates for our computations, which is relatively small. The decomposition into
factors allows us to take into full consideration not only the parallel shifts, but also other
additional factors, which push up the VaR number.
4.6
APPENDIX: PRINCIPAL COMPONENT ANALYSIS
The factors we used in the previous sections were, to some extent, ad hoc. We chose
to define the level factor as the average yield across maturities, the slope factor as the
difference in yields between the 10-year and 1-month bond, and finally, the curvature factor
as the butterfly spread. Of course, given that we defined these factors somewhat arbitrarily,
there is always the possibility that a better choice of factors exists. In particular, these
factors should satisfy some conditions in order to structure an effective risk management
strategy:
1. First, the factors ought to “explain” a very large proportion of the variation of the
yields of bonds at various horizons.
2. Second, the factors should be independent of each other.
The first condition is intuitive: If the factors do not explain the variation of interest rates,
they won’t be of very much use to a risk manager. The second condition is also intuitive:
If the level of interest rates moves inversely to its slope, for instance, then our hedging
strategy must take this joint movement into account. Instead, the factor neutrality discussed
in the previous section does not.
Principal component analysis (PCA) is a statistical technique that identifies the best
factors from historical yields, where the term best is in the sense of the two conditions
mentioned above. We now cover the basics of PCA, with a warning that this material is
more advanced than the material in previous sections, and it can be skipped by the less
mathematically sophisticated reader.
11 The
portfolio here is slightly different from the one in Section 3.8 of Chapter 3, as discussed earlier.
146
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
First, note that all the factors defined earlier are linear combinations of the underlying
yields ri (t). For instance, the level of interest rate was defined as the average of yields
n
across maturities, which implies it is given by φ1 = i=1 1/nr1 (t), where n is the number
of maturities. We will still impose this restriction, that is, for i = 1, 2, 3 we are going to
define factors as follows
φPi C A (t) = ai1 × r1 (t) + ... + ain × rn (t)
Denote the covariance between Δrk (t) and Δr (t) by
σ k = Cov (Δrk (t), Δr (t))
Principal component analysis determines the factors one at a time as follows. Consider
the first factor φP1 C A (t). The objective is to find coefficients a11 , ..., a1n to maximize the
variance of ΔφP1 C A (t):
max
α 1 1 ,...,α 1 n
V ar ΔφPi C A
n
n
=
a1k a1 σ k (4.36)
k =1 =1
under the restriction
n
a21j = 1.
(4.37)
j=1
Since ΔφP1 C A (t) is a linear combination of all the yields and it is obtained by maximizing its variability, it is intuitive that it will covary substantially with all the yields
Δr1 (t), ..., Δrn (t) . In a sense, it is as if we are maximizing some weighted average of the
R2 of the multiple regressions
Δr1 (t)
= α1 + β 11 ΔφP1 C A (t) + ε1 (t)
Δr2 (t)
= α2 + β 21 ΔφP1 C A (t) + ε2 (t)
..
.
Δrn (t)
.
= ..
(4.38)
= αn + β n 1 ΔφP1 C A (t) + εn (t)
Once we solve the maximization in Equation 4.36 and therefore estimate the first
component, we can compute the residual of the regressions in Equations 4.38: For all
i = 1, .., n we compute
εi (t) = Δri (t) − αi − β i1 ΔφP C A (t)
The residuals εi (t) provide the amount of variation in the various rates that the first factor
φP1 C A (t) is not able to explain. Hence, we can compute the change in the second factor
ΔφP2 C A (t) as
ε1 (t) + ... + a2n × εn (t)
ΔφP2 C A (t) = a21 × 147
APPENDIX: PRINCIPAL COMPONENT ANALYSIS
We find again the a21 , ..., a2n that maximize
max
a 2 1 ,...,a 2 n
V ar ΔφP2 C A (t)
(4.39)
conditional again on j = 1 a22j = 1. And so on.
The outcome of this methodology is to obtain factors that explain a large variation of
the cross-section of yields. Moreover, because at every step we are using the residuals of
the yields, the factors so constructed are independent of each other. That is, PCA generates
factors that satisfy conditions 1 and 2 above. The actual implementation of PCA is not as
difficult as it may appear from the methodology described above, and it is further described
in the next subsection. Here, instead, we focus on the outcome.
n
Figure 4.9
Coefficients on Level, Slope and Curvature from PCA
0.3
0.2
0.1
0
βij
−0.1
−0.2
−0.3
−0.4
Level
Slope
−0.5
Curvature
−0.6
0
1
2
3
4
5
6
Time to Maturity
7
8
9
10
Figure 4.9 shows the coefficients β Pij C A estimates from the PCA across maturity. The
solid line denotes the coefficients β Pi1C A that multiply the first factor coming from the
first maximization in Equation 4.36. The plot shows that all of these coefficients are
approximately equal to each other across maturities. Because all of the β Pi1C A are similar
to each other, the first factor computed from the PCA analysis has been called the level
factor: When this first factor φ1 (t) increases, all of the yields increase by about the same
amount. The level of β Pij C A is smaller than the one in Table 4.5 because in PCA we
constrain the squared coefficients to sum to one. This is simply a renormalization, as the
factor φ1 (t) becomes automatically scaled up to take this into account. One important
difference between this methodology and the one developed in the previous sections is the
fact that the level factor computed from PCA according to Equation 4.36 is a result of the
statistical methodology, and not an assumption as we made in Section 4.2. Panel A of
Figure 4.10 plots the level factor from PCA.
148
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
Figure 4.10
Level, Slope and Curvature Factors from PCA
Panel A. Level
Percent
40
20
0
1965
1970
1975
1980
1985
1990
1995
2000
2005
1990
1995
2000
2005
1990
1995
2000
2005
Panel B. Slope
Percent
10
5
0
−5
1965
1970
1975
1980
1985
Panel C. Curvature
Percent
2
0
−2
−4
−6
1965
1970
1975
1980
1985
The dotted line in Figure 4.9 shows the coefficients β i2 multiplying the second factor
φ2 (t). The second factor that comes out of the methodology described above is called
a slope factor exactly because the coefficients β i2 are negative for short-term maturities
and positive for long-term maturities. That is, when the second factor φ2 (t) increases, the
short-term yields decline and the long-term yields increase: i.e., the term structure becomes
steeper. The historical time series of the second factor obtained from PCA is plotted in
Panel B of Figure 4.10. Finally, the dashed line in Figure 4.9 shows the coefficients β Pi3C A
multiplying the third factor φ3 (t): The coefficients are negative for short- and long-term
yields, and positive for medium-term yields. That is, when the third factor increases, the
yield curve becomes more curved. For this reason, the third factor from PCA is called
curvature. The historical time series of the curvature factors is plotted in Panel C of Figure
4.10.
APPENDIX: PRINCIPAL COMPONENT ANALYSIS
149
Table 4.15 The Sensitivity of Interest Rates to PCA Level, Slope, and Curvature
Factor
3 months
6 months
1 year
Maturity
2 year
4 year
6 year
8 year
10 year
β Pi 1C A
R2
0.2637
75.50 %
0.2827
88.23%
0.2835
96.32%
Panel A: Level
0.2612
0.2172
95.89% 89.12%
0.1848
81.13%
0.1633
70.73%
0.1490
61.12%
β Pi 2C A
R2
−0.4017
98.26%
−0.2410
96.56%
−0.0807
97.33%
Panel B: Slope
0.0757
0.1983
96.94% 98.77%
0.2286
97.26%
0.2355
89.84%
0.2369
81.20%
β Pi 3C A
R2
−0.1679
99.49%
0.0672
96.76%
0.1994
99.25%
Panel C: Curvature
0.1899
0.0117
98.98% 98.78%
−0.1690
99.98%
−0.2996
99.40%
−0.3890
97.94%
The R 2 in Panels A, B, and C refer to the total R 2 from including Factor 1, Factors 1 and 2, and all three factors,
respectively.
4.6.1
Benefits from PCA
As mentioned before, and worth emphasizing, PCA has additional benefits over the more
intuitive methodology described earlier, namely, of defining the level, slope, and curvature
factors straight as yield average, term spread, and butterfly spread, respectively. These
additional benefits are the following:
1. The first additional benefit is that PCA explains an even greater fraction of the
variation of yields. This is shown in Table 4.15. This table contains additional
information compared to Figure 4.9. In particular, it shows the ability of each of the
factors to explain the variation in the interest rates, through the measure R2 . For
instance, the first PCA factor (level) explains 75% of the variation in the 3-month
yield, 95.89% of the variation of the 2-year rate, and 61.12% of the variation of the
10-year rate. The first and second factors together explain the largest part of the
variation in the term structure: The R2 of the rates up to 6 years are all above 96%.
The three factors together explain almost all of the variation, with R2 typically above
the 99%. It is important to note that these R2 s are always above the ones obtained
using the ad hoc factors in Table 4.5, showing that the factors obtained from PCA
are better at explaining the variation of yields, which is what a risk manager wants.
2. The second additional benefit is that the PCA factors are in fact independent from
each other. This implies that risk management practices can be more easily performed
using PCA factors than the ad hoc factors discussed earlier. We demonstrate this in
the next example.
EXAMPLE 4.8
In this example we perform the same exercise as in Example 4.7, but we use the PCA
factors to hedge against level and slope. The only difference from before is that we
must use now the parameters in Table 4.15 to compute both the factor durations of
the 3.875% bond (expiring on February 15, 2013) and the position in the short-term
150
BASIC REFINEMENTS IN INTEREST RATE RISK MANAGEMENT
and long-term bonds kS and kL . Note that the hedging formulas in Equations 4.29
and 4.30 are still valid even under the new methodology to compute the factors. In
particular, using again the zero coupon bonds maturing on February 15, 2005 and on
February 15, 2013, we find the following:
Long-term coupon bond
: D1 = 1.2162; D2 = 1.7240;
Short-term zero : D1S = 0.2473; D2S = −0.0704;
Long-term zero : D1L = 1.3793; D2L = 2.0864.
Using Equations 4.29 and 4.30 we obtain that the position kS and kL in the shortterm and long-term zero coupon bonds are
kS = −0.2669; kL = −1.1917
H edg e
In this case, on April 15 2007 the hedged portfolio is worth V4/15/2004
= $101.29,
i.e. a drop of only 0.20%, against the 0.50% using the ad hoc factors, and the 3.30%
using only duration.
4.6.2
The Implementation of PCA
In practice, the implementation of PCA is not as cumbersome as it seems, but it requires
the use of advanced computer packages. Start from the variance-covariance matrix of
Δri (t)’s, which we denote by M = Cov (Δri (t)). This is a n × n matrix. The properties
of this matrix are such that there are numbers λi and associated vectors
⎞
⎛
vi1
⎟
⎜
vi = ⎝ ... ⎠
vin
such that the following relation holds
Mvi =λi vi
(4.40)
The numbers λi are called eigenvalues and the associated vector vt is called eigenvector.
These eigenvalues and eigenvectors can be efficiently computed numerically through the
use of computers. Commercial packages are available for their computation.
Consider the maximization problem in Equations 4.36 and 4.39. Using vector notation
a1 = (a11 , ..., a1n ), we can form the Lagrangean as L(a1 ) = a1 Ma1 − λ (a1 a1 − 1). The
first order condition is
∂L(a1 )
= Ma1 − λa1 = 0
(4.41)
∂a1
which we recognize is satisfied by the eigenvalues λi and eigenvectors vi in Equation 4.40.
This result implies that, assuming we have available a computer package to compute the
eigenvalues and the eigenvectors of M , the procedure is then the following:
1. Compute the eigenvalues of M, denote them by λ1 > λ2 > ... > λn and let
v1 , v2 , ..., vn be their (normalized) eigenvectors.
APPENDIX: PRINCIPAL COMPONENT ANALYSIS
151
2. From Equation 4.41 v1 is the solution to the first maximization in Equation 4.36,
that is v1 = (a11 , ..., a1n )
3. Accordingly, we set
n
ΔφP1 C A (t) =
v1k Δrk (t)
k =1
εn (t)
4. We then run the regression in Equation 4.38 and compute the residuals ε1 (t) , .., 5. A similar argument as above shows that v2 is the solution to the second maximization
in Equation 4.39. Accordingly, we set the second factor
n
ΔφP2 C A (t) =
v2k εk (t)
k =1
6. We run the regression in Equation 4.33 with two factors. And so on
The result of this procedure are contained in Table 4.15 and Figures 4.9 and 4.10.
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CHAPTER 5
INTEREST RATE DERIVATIVES:
FORWARDS AND SWAPS
Interest rate “derivatives” play a central role in modern financial markets. The quotation
marks around the word “derivatives” is almost mandatory nowadays. Traditionally, we
think of a derivative security as a security whose value depends on the value of some other
more basic security. That is, the value of the derivative security “derives” from the one
of a primitive security. That’s the traditional view, which was fine at the time in which
basic derivative securities, such as forwards, futures, and swaps, were introduced in the
1970s and 1980s. Nowadays, however, the market size of interest rate derivatives is much
larger than the market of the primary securities. For instance, in this chapter we learn how
to compute the value of swap contracts from discount factors, possibly obtained from the
prices of Treasury securities, as discussed in Chapter 2. However, as we do so, we must
keep in mind that while as of December 2008 the market size of U.S. Treasury securities
was around $5.9 trillion, the global market value of swaps was about $16 trillion. The
obvious question is then whether the value of swaps depends on the value of Treasuries,
or vice versa. Here we are in a “chicken-and-egg” situation, because we cannot be sure
anymore which one is the primary market and which are the “derivative markets.” In this
chapter we learn that some relations – no arbitrage relations – must exist between the values
of the securities. If the relative values between Treasuries and derivatives get out of line,
then speculators will step in and correct the market imbalance. We should think of all of
these markets as moving in sync, and when one moves because of its own forces, the others
– whether derivatives or primary – should adjust accordingly.
153
154
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Figure 5.1 A Forward Investment Need
Firm receives
$100 million
receivable
Desired investment period at
rate f locked in at 0
2
Today = 0 = Mar 1, 2001
T = 0.5 = Sep 3, 2001
1
T = 1 = Mar 1, 2002
2
Treasury bill prices
P (0,0.5) = $97.728
bill
P (0 , 1) = $95.713
bill
5.1 FORWARD RATES AND FORWARD DISCOUNT FACTORS
It is convenient to introduce the concept of forward rate through an example:
EXAMPLE 5.1
Let today be March 1, 2001.1 Suppose a firm sold a piece of equipment to a client
for $100 million. The client will pay in six months, i.e. on T1 = September 4,
2001 (the first business day in September 2001). Suppose the firm does not need
that cash immediately, but it will need it six months later, at T2 = March 1, 2002,
to fund some capital investment. Today, the firm would like to fix the interest rate
to be applied on the receivable of $100 million for the six month period from T1
to T2 . Figure 5.1 illustrates the investment need. The firm calls up its bank to ask
for a quote, and the bank quotes the (semi-annually compounded) annualized rate of
f2 = 4.21%.2 That is, the bank is ready to commit today to receive in six months
(at T1 ) $100 million from the firm, and return six months later (at T2 ) the amount
$102.105 = $100 × (1 + f2 /2) million. The rate, f2 , which the bank commits to
today is called forward rate.
1 The
date 2001 may seem peculiar. This choice was driven by the fact that in March, 2001 the U.S. Treasury
still issued 1-year Treasury bills, a practice that it discontinued shortly thereafter until recently. The presence
of 1-year Treasury bills makes the example easier to follow, although the same argument can be repeated using
Treasury STRIPS, or synthetically produced zero coupon bonds.
2 The subscript “2” in f denotes semi-annually compounding, as the notation used for interest rates. See Equation
2
5.3 below for details.
FORWARD RATES AND FORWARD DISCOUNT FACTORS
Table 5.1
Trading Strategy to Compute Forward Rate
T1
Today (Time 0)
Sell short $97.728 m of T-bills
maturing at T1
T2
(a) Receive $100 m
from firm
(b) Close short position
Buy M = 1.02105 = $$ 99 75 .. 77 21 83
of T-bills maturing at T2
Total Net Cash Flow = 0
155
(a) Receive 1.02105 × $100 m
(b) Give total to firm
Total Net Cash Flow = 0
Total Net Cash Flow = 0
The question is: How does the bank determine the forward rate f2 ?
By no arbitrage. In fact, on March 1, 2001 (today), the value of 6-months
Treasury bills is $97.728 and the value of 1-year Treasury bills is $95.713. In order
to guarantee the rate f2 to the client, the bank can perform the following strategy (see
Table 5.1): First, today the bank borrows T-bills with maturity T1 = 6 months and
sells them for $97.728 million. This amount of cash is then invested in 1-year T-bills
expiring in T2 . Given the price of the latter of $95.713 the bank can now purchase
M = 1.02105 = $97.728/$95.713 million of 1-year T-bills (for $100 of face value).
Today, the net cash flow remaining to the bank is zero, as all the cash obtained from
the sale of the T1 T-bill has been used to purchase T2 T-bills.
Second, at time T1 the bank must pay back $100 million to the counterparty it
borrowed the T1 T-bills from. However, this is also the time in which the firm will give
the bank $100 million. So, also at T1 the net cash flow is zero, as the bank receives
$100 million from the firm and uses it to close the short position. Finally, at time
T2 the M = 1.02105 T-bills mature, and the bank receives M × $100 = $102.105
million. This is exactly the cash flow that the bank promised the firm in return on the
investment of $100 million at T1 .
Indeed, the return for the firm from T1 to T2 is 2.105% = ($102.105−$100)/$100,
which implies the annualized interest rate of f2 = 2.105% × 2 = 4.21%.
This example shows that the term structure of interest rates, or equivalently, the discount
factors Z(0, T ), contain all the information needed to establish the time value of money
in the future. In the example, the current 1-year discount is Z(0, 1) = 0.95713, while the
6-month discount is Z(0, 0.5) = 0.97728. That is, $1 in one year is worth 95.71 cents in
today’s money, while $1 in six months is worth 97.73 cents in today’s money. Implicitly,
these two exchange rates (of money in the future for money today) imply an exchange rate
of money in six months versus money in one year, given by the ratio of the two discounts
F (0, 0.5, 1) =
Z(0, 1)
= 0.97938
Z(0, 0.5)
(5.1)
That is, given the information today (time 0), $1 in one year is equivalent to 97.93 cents in
six months. Given the two values of Z(0, 0.5) and Z(0, 1), it cannot be otherwise without
generating an arbitrage opportunity. Indeed, the reasoning behind the computation of the
156
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
time value of money between future dates implied by the current term structure of interest
rates is the same that applies to foreign exchange rates: If 1 U.S. dollar is worth 1.25
euros and 1 dollar is worth 1.6 British pounds, the exchange rate between euros and British
pounds must be 0.78=1.25/1.6.
One way to interpret F (0, 0.5, 1) determined in Equation 5.1 is as the current market
projection of the future discount factor Z(0.5, 1), which is not known today. For this
reason, we call F the forward discount factor.
Definition 5.1 The forward discount factor at time t defines the time value of money
between two future dates, T1 and T2 > T1 . Given the discount factors Z(t, T1 ) and
Z(t, T2 ), the forward discount factor is given by
F (t, T1 , T2 ) =
Z(t, T2 )
Z(t, T1 )
(5.2)
From its definition in Equation 5.2, F (t, T1 , T2 ) has the standard properties of a discount
factor, as discussed next.
Fact 5.1 The forward discount factor has the following properties:
1. F (t, T1 , T2 ) = 1 for T2 = T1 ;
2. F (t, T1 , T2 ) is decreasing in T2 .
The second property stems from Equation 5.2: As we increase T2 , only the numerator
in the formula changes, and this is decreasing with T2 . Thus, F (t, T1 , T2 ) decreases with
T2 .
Going back to the previous example, because F (0, 0.5, 1) is a discount factor between
times T1 = 0.5 and T2 = 1, we can also compute its implied semi-annually compounded
interest rate by using the same formula as in Equation 2.4 in Chapter 2, in which Z(t, T )
is substituted for F (0, 0.5, 1):
f2 (0, 0.5, 1) = 2 ×
1
1
F (0, 0.5, 1) 2 ×0 . 5
−1
= 4.21%,
which is what we found earlier in Example 5.1. Indeed, we have the following definition:
Definition 5.2 The forward rate at time t for a risk free investment from T1 to T2 , and with
compounding frequency n, is the interest rate determined by the forward discount factor in
Equation 5.2,
1
−1
(5.3)
fn (t, T1 , T2 ) = n ×
1
F (t, T1 , T2 ) n ×( T 2 −T 1 )
The continuously compounded forward rate is obtained for n that grows to infinity:
f (t, T1 , T2 ) = −
ln (F (t, T1 , T2 ))
T2 − T1
(5.4)
FORWARD RATES AND FORWARD DISCOUNT FACTORS
157
As it is true with spot rate and discount factors, there is an equivalence between forward
rates and forward discount factors: Given a forward discount factor, we can determine
a forward rate from Equations 5.3 or 5.4. Conversely, given a forward rate with its
compounding frequency, we can determine the forward discount factor as follows:
Fact 5.2 Given a n-times compounded forward rate fn (t, T1 , T2 ), the forward discount
factor is
F (t, T1 , T2 ) = 1
1+
f n (t,T 1 ,T 2 )
n
n ×(T 2 −T 1 )
(5.5)
Given a continously compounded forward rate f (t, T1 , T2 ), the forward discount factor is
F (t, T1 , T2 ) = e−f (t,T 1 ,T 2 )(T 2 −T 1 )
(5.6)
To conclude this section, note that if for any two dates T1 and T2 > T1 the discount
factor Z(t, T ) is increasing, i.e. Z(t, T1 ) < Z(t, T2 ), then from Equation 5.2 the forward
discount factor F (t, T1 , T2 ) > 1, which in turn implies that both fn (t, T1 , T2 ) < 0 and
f (t, T1 , T2 ) < 0. That is:
Fact 5.3 If the discount factor Z(t, T ) is increasing between two dates T1 and T2 > T1 ,
that is, Z(t, T1 ) < Z(t, T2 ), then the forward rate at t for an investment between T1 and
T2 is negative.
In the context of Example 5.1, a situation in which Z(0, T1 ) < Z(0, T2 ) would imply
that the bank would be willing to quote a negative interest rate f2 to the firm for the future
investment between T1 and T2 , which is not reasonable. The discount factors Z(t, T )
must therefore be decreasing with maturity T , as we already noticed in Section 2.1.1 in
Chapter 2.
5.1.1
Forward Rates by No Arbitrage
The argument used in Example 5.1 to establish the forward rate f2 that the bank will apply
to the firm’s investment at time T1 , as agreed upon today, is a no arbitrage argument. It may
prove useful to view it again, more generally, as follows. Suppose that at time t we need
to invest some amount of money $W until a future date T2 . Consider any intermediate
date T1 , and assume that risk free zero coupon bonds are available for an investment from
t to T1 and for an investment from t to T2 . As we know, the prices of these zero coupon
bonds are Pz (t, T1 ) = 100 × Z(t, T1 ) and Pz (t, T2 ) = 100 × Z(t, T2 ).3 The following
two possible strategies are available to us:
3 As
discussed in Chapter 1 zero coupon bonds for long maturities are available for purchase in U.S. through the
STRIPS program.
158
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Strategy 1. Invest $W in T2 zero coupon bonds. Since we can purchase $W/Pz (t, T2 )
zero coupon bonds and the zero coupon bonds pay $100 at T2 , the payoff at T2 is
$W
Payoff at T2 of strategy 1 =
× $100
(5.7)
Pz (t, T2 )
Strategy 2. Invest $W in T1 zero coupon bonds, and agree with a bank on a semiannually compounded forward rate f2 to be applied for an investment from T1 to T2 .
Since we can purchase $W/Pz (t, T1 ) zero coupon bonds and the zero coupon bonds
pay $100 at T1 , the payoff at T2 is
$W
2×(T 2 −T 1 )
(5.8)
Payoff at T2 of strategy 2 =
× $100 ×(1 + f2 /2)
Pz (t, T1 )
Because under either strategy the investor obtains an amount of money at time T2 that
is known today, and thus without any risk, the two strategies must yield the same payoff,
otherwise an arbitrage opportunity arises. For instance, if strategy 2 yielded a higher payoff
than strategy 1, then anybody holding T2 bonds would sell them to purchase T1 bonds and
would at the same time enter into agreements with banks for a forward investment at the
rate f2 . This strategy would push down the price of T2 bonds and rise the price of T1 bonds,
as well as forward rates f2 , reequilibrating financial markets.
In order for markets to be in equilibrium, we must have
Payoff at T2 of strategy 2 = Payoff at T2 of strategy 1
or, substituting for the expressions in Equations 5.7 and 5.8:
$W
$W
2×(T 2 −T 1 )
=
× $100 × (1 + f2 /2)
× $100
Pz (t, T1 )
Pz (t, T2 )
(5.9)
(5.10)
From this equation, we can substitute for the prices of zero coupon bonds Pz (t, T1 ) =
100 × Z(t, T1 ) and Pz (t, T2 ) = 100 × Z(t, T2 ), and then simplify from both sides $W
and $100. As an intermediate step, we can write
(1 + f2 /2)
2×(T 2 −T 1 )
=
Z(t, T1 )
Z(t, T2 )
(5.11)
Comparing with Equation 5.2, the right hand side of this last equation is the reciprocal of
the forward discount factor Z(t, T1 )/Z(t, T2 ) = 1/F (t, T1 , T2 ). Solving for f2 , we indeed
obtain Equation 5.3 for n = 2
f2 = f2 (t, T1 , T2 ) = 2 ×
1
1
F (t, T1 , T2 ) 2 ×( T 2 −T 1 )
−1
(5.12)
5.1.2 The Forward Curve
In the case of forward rates and forward discount factors, we have three time indices: t,
which is the time when we decide we want to lock in the future interest rate, and then the
two future dates T1 and T2 during which the investment will take place. In this section, it
FORWARD RATES AND FORWARD DISCOUNT FACTORS
159
is convenient to set t = 0, and denote the first of the two future dates by T . Finally, assume
that the future investment (at T ) will be for a period of length Δ. In addition, and again
for convenience, we focus in this section on the continuously compounded forward rate,
which can then be denoted by
f (0, T, T + Δ) = −
ln (F (0, T, T + Δ))
Δ
(5.13)
Keeping fixed Δ, we can plot the forward rate f (0, T, T + Δ) with respect T . The
resulting curve is called forward curve.
Definition 5.3 The forward curve gives the relation between the forward rate f (0, T, T +
Δ) and the time of the investment T .
For example, Figure 5.2 plots the forward curve f (0, T, T + 0.25) for T that varies from
three months to ten years.4 The figure also plots the underlying spot curve r(0, T ) for the
same period. The forward curve and spot curve seem to move in tandem: Roughly, they go
up and down at the same time. In fact, there is a precise relation between forward curves
and spot curves, as, remember, the forward curve is derived from the same discount factors
that determine the spot curve.
Figure 5.2
The Spot Curve and the Forward Curve
6.7
Spot Curve
Forward Curve
6.6
6.5
Interest Rate (%)
6.4
6.3
6.2
6.1
6
5.9
5.8
5.7
0
1
2
3
4
5
6
Time to Maturity
7
8
9
10
Data Source: CRSP.
4 Both
the yield curve and the forward curve are calculated based on data from CRSP (insert database name)
c 2009 Center for Research in Security Prices (CRSP), The University of Chicago Booth School of Business.
160
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Indeed, by definition of the forward discount factor, we find:
F (0, T, T + Δ)
=
Z(0, T + Δ)
Z(0, T )
=
e−r (0,T + Δ )(T + Δ )
e−r (0,T )T
−r (0,T + Δ )(T + Δ )+r (0,T )T
e
=
e−r (0,T )Δ −(r (0,T + Δ )−r (0,T ))(T + Δ )
=
Substituting now into Equation 5.13 we obtain
r(0, T + Δ) − r(0, T )
(5.14)
Δ
This equation shows the key relation between forward rates and spot rates. For any
maturity T , the forward rate between T and T + Δ equals the spot rate with maturity T
plus a term that is positive if the spot curve is rising, and it is negative if the spot curve is
declining at T . To see this last point, note that the numerator of the last term in Equation
5.14 is the difference in the spot curve between the two maturities T and T + Δ. If this
difference is positive and Δ is sufficiently small, then the term structure is increasing at
T . If that difference is instead negative, the spot curve is decreasing at T . This formula
implies the following:
f (0, T, T + Δ) = r(0, T ) + (T + Δ) ×
Fact 5.4 When the spot curve is increasing, the forward curve is above the spot curve.
When the spot curve is decreasing, the forward curve is below the spot curve. When the
spot curve is flat, the forward curve equals the spot curve.
The pattern described in Fact 5.4 is apparent in Figure 5.2.
To better understand the relation that exists between forward rates f (0, T, T + Δ)
and the underlying spot curve r(0, T ), consider now a set of dates, T1 , T2 ,..., Tn , with
Ti+1 = Ti + Δ. Let also T1 = Δ. For each pair of dates, we can rewrite the forward
discount (Equation 5.2) as
Z(0, Ti+ 1 ) = Z(0, Ti ) × F (t, Ti , Ti+1 )
(5.15)
Using also Equation 5.4, the following relations hold
Z(0, T1 ) = e−r (0,T 1 )×Δ
(5.16)
−f (0,T 1 ,T 2 )×Δ
Z(0, T2 ) = Z(0, T1 ) × e
..
.
. = ..
Z(0, Ti ) = Z(0, Ti−1 ) × e−f (0,T i −1 ,T i )×Δ
..
.
. = ..
(5.17)
Z(0, Tn ) = Z(0, Tn −1 ) × e−f (0,T n −1 ,T n )×Δ
(5.18)
By substituting Z(0, T1 ) from the first equation into the second, and Z(0, T2 ) from the
second into the third, and so on up to Tn , we find that an alternative expression of Z(0, Tn )
is
FORWARD RATES AND FORWARD DISCOUNT FACTORS
Z(0, Tn ) = e−(r (0,T 1 )×Δ + f (0,T 1 ,T 2 )×Δ + ...+f (0,T n −1 ,T n )×Δ )
161
(5.19)
−r (0,T n )×T n
By definition of the continuously compounded yield r(0, Tn ), Z(0, Tn ) = e
we must have
,
r(0, Tn ) × Tn = (r(0, T1 ) × Δ + f (0, T1 , T2 ) × Δ + ... + f (0, Tn −1 , Tn ) × Δ) (5.20)
This last relation yields the following:
Fact 5.5 The continuously compounded spot rate r(0, Tn ) is equal to the average of
forward rates up to Tn
r(0, Tn ) =
1
Tn
n
f (0, Ti−1 , Ti ) × Δ
(5.21)
i= 1
where for convenience, we denote T0 = 0 and f (0, 0, T1 ) = r(0, T1 ).
Since the spot rate with maturity T is an average of forward rates, it becomes quite
intuitive that if the forward rate is above the spot rate, then the spot rate must be increasing
for the next step, as the incremental rate added to the average is above the average, pushing
the average up. Similarly, if the forward rate is below the spot rate with the same maturity,
the spot rate curve must decline in the next step, as the new additional element in the
average is below the average.
5.1.3
Extracting the Spot Rate Curve from Forward Rates
Sometimes we do not have available zero coupon bond prices to compute the discount
factors Z(0, T ). However, we may have data of forward rates f (0, Ti , Ti+1 ), for a set of
maturities Ti , i = 1, ..., n with Ti+ 1 = Ti + Δ, where Δ is an interval of time. Because
from these forward rates we can compute the forward discount factors F (0, Ti , Ti+1 ) (see
Equation 5.6), we can then use Equation 5.15 to extract the zero coupon yield curve. Essentially, this methodology amounts to using Equations 5.16 - 5.18 in reverse, from forwards
to discount factors Z(0, Ti ). The next example illustrates this bootstrap methodology.
EXAMPLE 5.2
Let today be May 5, 2008. The third column in Table 5.2 contains continuously
compounded forward rates at the quarterly frequency (Δ = 0.25).5 Note that the
first forward rate corresponds to the spot rate itself f (0, 0, 0.25) = r(0, 0.25). The
first entry in the third column determines the first discount
Z(0, 0.25) = e−f (0,0,0.25)×0.25 = e−2.7605% ×0.25 = 0.993123
which is reported in Column 4. Next, we compute the discount for the maturity
T2 = 0.5. Using Equations 5.6 and 5.15 we obtain:
Z(0, 0.5)
= Z(0, 0.25) × e−f (0,0.25,0.5)×0.25
= 0.993123 × e−2.9460% ×0.25
= 0.986521
5 Section
6.5 in Chapter 6 describes the source of these data.
162
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Table 5.2
Forward Rates and Spot Curve on May 5, 2008
Step
i
Maturity
Ti
(c.c.) Forward Rate
f (0, Ti −1 , Ti ) (%)
Discount
Z(0, Ti ) (×100)
(c.c.) Spot Rate
r(0, Ti ) (%)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
0.2500
0.5000
0.7500
1.0000
1.2500
1.5000
1.7500
2.0000
2.2500
2.5000
2.7500
3.0000
3.2500
3.5000
3.7500
4.0000
4.2500
4.5000
4.7500
2.7605
2.6677
2.8132
3.0088
3.1917
3.3664
3.5501
3.7118
3.8440
3.9700
4.0899
4.1800
4.2474
4.3150
4.3884
4.4466
4.4913
4.5395
4.5953
99.3123
98.6521
97.9607
97.2267
96.4539
95.6456
94.8005
93.9248
93.0265
92.1078
91.1708
90.2230
89.2701
88.3123
87.3487
86.3830
85.4186
84.4546
83.4900
2.7605
2.7141
2.7471
2.8125
2.8884
2.9680
3.0512
3.1338
3.2127
3.2884
3.3613
3.4295
3.4924
3.5512
3.6070
3.6595
3.7084
3.7546
3.7988
Original Data Source: Bloomberg
again reported in Column 4. We can proceed for every i = 2, ..., 19 and use the
formula
Z(0, Ti ) = Z(0, Ti−1 ) × e−f (0,T i −1 ,T i )×0.25
Column 4 contains the resulting discount factors Z(0, Ti ) for every maturity. Column
5 uses these discount factors to obtain the continuously compounded spot rate
r(0, Ti ) = −
5.2
ln(Z(0, Ti ))
Ti
FORWARD RATE AGREEMENTS
Definition 5.4 A Forward Rate Agreement (FRA) is a contract between two counterparties, according to which one counterparty agrees to pay the forward rate fn (0, T1 , T2 ) on a
given notional amount N during a given future period of time from T1 to T2 = T1 +Δ, while
the other counterparty agrees to pay according to future market floating rate rn (T1 , T2 ).
The net payment between the two counterparties at the maturity T2 of the contract is then
given by
(5.22)
Net payment at T2 = N × Δ × [rn (T1 , T2 ) − fn (0, T1 , T2 )]
FORWARD RATE AGREEMENTS
163
Above, Δ = T2 − T1 , typically a quarter or six months, while the subscript n = 1/Δ
denotes the corresponding compounding frequency, e.g., n = 4 or n = 2 for quarterly or
semi-annual compounding.
To understand the logic of FRAs, it is useful to go back to Example 5.1.
EXAMPLE 5.3
Recall from Example 5.1 that today (t = 0) is March 1, 2001, that a firm has a
receivable of $100 million in six months (T1 = 0.5), and the firm wishes to invest
this amount for an additional six months (until T2 = 1). An alternative strategy to
the one described in Example 5.1 is for the firm to enter into a six-month FRA with a
bank for the period T1 to T2 , and notional N = $100 million. That is, today the bank
agrees to pay in one year (T2 = 1) the amount N × f2 (0, 0.5, 1), where f2 (0, 0.5, 1)
is the current semi-annually compounded forward rate for the period T1 to T2 , while
the firm agrees to pay on the same day the amount N × r2 (0.5, 1), where r2 (0.5, 1)
is the semi-annually compounded spot interest rate at time T1 = 0.5. That is, they
exchange the payment at T2 = 1
N
× [f2 (0, 0.5, 1) − r2 (0.5, 1)]
(5.23)
2
where we must divide by two because the interest is only over six months. Recall
f2 (0, 0.5, 1) = 4.21%. We now see that the firm reaches exactly the same outcome
as in Example 5.1.
Indeed, at T1 = 0.5 when the firm receives its $100 million receivable, the firm
can simply invest this amount at the market interest rate r2 (0.5, 1). How much money
will the firm have at time T2 = 1? At this time, the firm receives the payoff from the
investment, plus the net payment given in equation (5.23). In total
r2 (0.5, 1)
Total amount
=
$100 million × 1 +
(Return on investment)
at T2
2
N
× [f2 (0, .5, 1) − r2 (0.5, 1)]
(FRA payment)
+
2
f2 (0, .5, 1)
= $100 million × 1 +
2
= $102.105 million
Net payment of the firm at T2 =
The firm is in exactly the same position as in Example 5.1.
What about the bank? In particular, the bank now is exposed to interest rate risk,
as the FRA yields a negative payoff if f2 (0, 0.5, 1) > r2 (0.5, 1). However, a little
modification to the strategy in Table 5.1 ensures the bank is hedged as well. The new
modified strategy is shown in Table 5.3. At time 0, the bank strategy is the same: The
bank shorts T1 T-bills and purchases M = 1.02105 T2 T-bills. At time T1 the bank
must come up with $100 million to pay the short position. The bank can borrow this
amount of money, at the current rate r2 (0.5, 1). At time T2 , the bank total cash flows
are as follows
r2 (0.5, 1)
Total CF
=
−$100 m × 1 +
(Pay back loan)
of bank at T2
2
164
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Table 5.3 Trading Strategy to Compute Forward Rate
T1
Today (Time 0)
Sell short $97.728 m of T-bills
maturing at T1
T2
(a) Borrow $100 m
at rate r2 (0.5, 1)
(b) Close short position
Buy M = 1.02105 = $$ 99 75 .. 77 21 83
of T-bills maturing at T2
r 2 (0 . 5 , 1 )
2
Receive 1.02105 × $100 m
Enter FRA with Firm
Total Net Cash Flow = 0
Pay $100 m × 1 +
Pay
Total Net Cash Flow = 0
100 m
2
× [f2 (0, 0.5, 1) − r2 (0.5, 1)]
Total Net Cash Flow = 0
+ {1.02105 × $100 m }
(T2 T-bills mature)
$100 m
+
× [r2 (0.5, 1) − f2 (0, 0.5, 1)]
(FRA payment)
2
= 0
A perfect hedge.
5.2.1
The Value of a Forward Rate Agreement
When two counterparties enter into a FRA, there is no exchange of money at the time of
the contract inception (time 0). In other words, the value of the forward rate agreement
is zero at inception. However, as time passes and forward rates change, the value of the
forward rate agreement changes as well. The following example demonstrates the issue.
EXAMPLE 5.4
Consider again Example 5.3 and suppose that three months after the initiation of the
contract, on June 1, 2001, the firm decides to close its FRA with the bank. So far,
the two counterparties have not exchanged any money, so it may appear that the firm
could simply call the bank and ask to close the contract. But as interest rates changed
between March 1 and June 1, so did the value of the forward rate agreement. To see
this, from Table 5.3 we note that at initiation (today = 0) the bank sold one T-bill
maturing at T1 and bought M = 1.02105 T-bills maturing at T2 . This portfolio (long
T2 T-bills and short T1 T-bills) exactly hedges the bank commitment to the FRA, as
discussed in Example 5.3. Since it produces exactly the cash flow that the firm will
receive, the value of this portfolio must reflect the value of the FRA for the firm.
Thus, for every t ≤ T1 , we have
Value of FRA to the firm at t = V F R A (t) = M × Pbill (t, T2 ) − Pbill (t, T1 )
where, recall M = Pbill (0, T1 )/Pbill (0, T2 ). For instance, at initiation we have
V F R A (0)
= M × Pbill (0, T2 ) − Pbill (0, T1 )
(5.24)
FORWARD RATE AGREEMENTS
=
=
165
Pbill (0, T1 )
× Pbill (0, T2 ) − Pbill (0, T1 )
Pbill (0, T2 )
0
showing that there is no exchange of money at initiation.
On June 1, 2001 (= t), the T1 T-bill price was Pbill (t, T1 ) = $99.10 and the T2
T-bill price was Pbill (t, T2 ) = $97.37. Therefore
V F R A (t)
=
1.02105 × Pbill (t, T2 ) − Pbill (t, T1 ) = 1.02105 × $97.37 − $99.1
= $0.319638 million
On June 1, 2001, the value of the forward rate agreement initiated 3 months earlier
was worth $319,638 to the firm. Simply calling up the bank and exiting from the
FRA would be a mistake, as the firm would lose this amount of money. Of course,
knowing this, the bank would pay this sum of money to the firm in case it wanted to
exit from the contract, minus some transaction costs.
More generally, we can compute the value of the FRA by considering separately the
payments of the two counterparties. In particular, let’s first decompose the payment of the
FRA as follows
Net payment at T2
=
N × Δ × [fn (0, T1 , T2 ) − rn (T1 , T2 )]
= N × [1 + fn (0, T1 , T2 )Δ] − N × [1 + rn (T1 , T2 )Δ]
= Fixed leg payment − Floating leg payment
We can then value the two legs separately. The fixed leg is the simplest, as this
corresponds to a fixed payment at time T2 and thus we can discount it, as if it was a zero
coupon bond:
Value of fixed leg at t
V f ixed (t)
= Present value of N × [1 + fn (0, T1 , T2 )Δ]
= Z(t, T2 ) × N × [1 + fn (0, T1 , T2 )Δ]
The value of the floating leg requires a little trick, as we do not know today (t) what the
payment will be at time T2 . First, we compute the value of the floating leg at T1 :
Value of floating leg atT1
V f loatin g (T1 )
= Present value of N × [1 + rn (T1 , T2 )Δ]
=
Z(T1 , T2 ) × N × [1 + rn (T1 , T2 )Δ]
1
× N × [1 + rn (T1 , T2 )Δ]
=
[1 + rn (T1 , T2 )Δ]
= N
Because the floating leg at T1 is always equal to N independently of the floating rate
rn (T1 , T2 ), we then find
Value of floating leg at t
V f loatin g (t)
= Present value of V f loatin g (T1 )
= Z(t, T1 ) × N
166
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Combining the value of the floating leg with the value of the fixed leg, we obtain
Fact 5.6 The value at time t of a forward rate agreement in which counterparties exchange
at T2 the cash flow
Net payment at T2 = N × Δ × [fn (0, T1 , T2 ) − rn (T1 , T2 )]
is given by
Value of FRA at t = V F R A (t)
= V f ixed (t) − V f loatin g (t)
= N × [M × Z(t, T2 ) − Z(t, T1 )]
where
M = 1 + fn (0, T1 , T2 )Δ =
(5.25)
Z(0, T1 )
Z(0, T2 )
The definition of M ensures that indeed the value of the FRA is zero at initiation
V F R A (0) = N × [M × Z(0, T2 ) − Z(0, T1 )] = 0
We can rewrite the value of the FRA in Equation 5.25 in a more intuitive way: Factor
out Z(t, T2 ) and obtain the following:
Fact 5.7 The value of a FRA can be expressed equivalently as follows:
Z(t, T1 )
F RA
V
(t) = N × Z(t, T2 ) × M −
Z(t, T2 )
= N × Z(t, T2 ) × Δ × [fn (0, T1 , T2 ) − fn (t, T1 , T2 )]
(5.26)
as we recall that by definition of a forward rate we have 1+fn (t, T1 , T2 )Δ = Z(t, T1 )/Z(t, T2 ).
Equation 5.26 is intuitive, and its logic is better explained by going back to Example 5.4.
EXAMPLE 5.5
In Example 5.4 the firm would like to exit the FRA 3 months after inception, call
it time t. Instead of calling up the counterparty and asking to close the contract,
the firm can achieve the same result by entering into a new FRA at time t with the
reversed payoff, that is
N
× [r2 (T1 , T2 ) − f2 (t, T1 , T2 )]
(5.27)
2
Clearly, the payoff of the FRA at time t depends on the current forward rate
f2 (t, T1 , T2 ) instead of the old one f (0, T1 , T2 ). The total payoff for the firm at T2
is then
Payoff of reverse FRA at T2 =
Total payoff at T2 of old FRA
+ new FRA
=
N
× [f2 (0, T1 , T2 ) − r2 (T1 , T2 )]
2
167
FORWARD CONTRACTS
+
=
N
× [r2 (T1 , T2 ) − f2 (t, T1 , T2 )]
2
N
[f2 (0, T1 , T2 ) − f2 (t, T1 , T2 )]
2
That is, at the time of the decision to close the original FRA contract, the firm will
end up with a positive payoff if the current forward rate declined since inception,
and vice versa. Since Pbill (t, T1 ) = $99.10 and Pbill (t, T2 ) = $97.37, the current
forward is f2 (t, T1 , T2 ) = 2 × (.991/.9737 − 1) = 3.55% < f (0, T1 , T2 ) = 4.21%.
The time T2 payoff is then N/2 [f2 (0, T1 , T2 ) − f2 (t, T1 , T2 )] = $328, 272, which
is known at the earlier time t. Therefore, the present value is
V F R A (t)
N
× [f2 (0, T1 , T2 ) − f2 (t, T1 , T2 )]
2
0.9737 × $328, 272 = $319, 638
= Z(t, T2 ) ×
=
as we obtained earlier.
5.3
FORWARD CONTRACTS
In a forward rate agreement, two counterparties agree to exchange cash flows according to
the difference between the forward rate (known at initiation of the contract) and the future
spot rate. An equivalent strategy for an investor to lock in a given rate of return in the future
is to agree to purchase a given Treasury security in the future, at a price determined today.
Consider again Example 5.1.
EXAMPLE 5.6
On March 1, 2001 (today = time 0), the firm may enter into a forward contract with
a bank to purchase six months later (T1 = 0.5) $100 million-worth of 6-months
Treasury bills for a price P f w d , for $100 par value, specified today.
What purchase price would the bank quote to the firm for the 6-month T-bills?
The answer is simply the forward discount factor F (0, .5, 1) computed in Equation
5.1, multiplied by 100. That is
Forward price = P f w d = 100 × F (0, .5, 1) = $97.938
(5.28)
To see why, go back to Table 5.1, and consider the bank’s trading strategy at time
0. Recall that from the (short) sale of T1 T-bills, the bank purchases M = 1.02105
million of T2 T-bills. Recall also that the net cash flow to the bank at time 0 is zero.
At time T1 the bank has to cover the short position, and uses the $100 million from the
firm, as in Table 5.1. Note that at this point the bank holds an amount M = 1.02105
million of T2 T-bills, which now have maturity T2 − T1 = six months. Therefore,
the bank can use these M 6-month T-bills to honor the terms of the forward contract.
That is, at T1 the bank simply delivers its own holdings M of 6-month T-bills to the
firm. This number M of T2 T-bills is exactly the number of 6-month T-bills that are
168
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
needed to ensure the firm gets $100 million-worth of 6-month T-bills, as the firm
requested. In fact, given the forward price P f w d in Equation 5.28, we have
$100 million
= 1.02105 million = M
P f wd
This example leads to the following definition:
Definition 5.5 A forward contract is a contract between two counterparties in which one
counterparty agrees to purchase and the other agrees to sell a given security at a given
future time, and at a given price, called the forward price. Denoting by P f w d (0, T, T ∗ )
the forward price at time 0 for delivery at time T of a security expiring at T ∗ , whose value
at T we denote by P (T, T ∗ ), the payoff at T of the forward contract is
Payoff from long forward contract = P (T, T ∗ ) − P f w d (0, T, T ∗ )
(5.29)
The payoff from being short the forward contract is the negative of Equation 5.29.
In Example 5.6 the underlying security was a 6-month Treasury Bill: The forward
contract yields the payoff in Equation 5.29 at time T because the firm receives a security
worth P (T, T ∗ ) but it pays P f w d (0, T, T ∗ ).
The next example illustrates that hedging with the forward contract yields the same
return on investment as the hedging strategy performed with forward rate agreements.
EXAMPLE 5.7
Consider again Example 5.6, and let P f w d (0, 0.5, 1) = 100×F (0, 0.5, 1) = $97.938
be the forward price quoted at time 0 for the investment between T1 = 0.5 to T2 = 1,
as determined in the example. Let the firm enter into a forward contract to purchase
M = 1.02105 million of 6-month T-bills on T1 = 0.5 (with $100 principal). The
payoff from the forward contract at time T1 = 0.5 is then given by
Payoff forward contract at T1 = M × (Pbill (T1 , T2 ) − $97.938)
Ex post, the fear of the firm that the interest rate would decline is in fact realized,
and the price of the 6-month T-bill at T1 turns out to be Pbill (T1 , T2 ) = $98.89 >
$97.938. Thus, the payoff from the forward contract is:
Payoff forward contract at T1 = 1.02105 million × ($98.89 − $97.938) = $972, 043.54
The firm at T1 can then invest this additional amount, $972,043.54, together with the
receivable $100, 000, 000 into the new T-bill. In particular, it will be purchasing an
amount of T-bills equal to
Investment in T-bills at T1 =
($100, 000, 000 + $972, 043.54)
= 1, 021, 054.136
$98.89
where the T-bills have $100 principal. At maturity T2 the total amount realized is
then $102,105,413.6, which implies a realized annualized rate of return equal to
Payoff at T2
1
×
−1
Annualized rate of return =
T2 − T1
Investment at T1
FORWARD CONTRACTS
1,021,054.136
−1
2×
100,000,000
4.21%
169
=
=
(5.30)
This is exactly equal to the forward rate determined in Example 5.1 and is in fact
independent of the realized Treasury bill T1 = 0.5, Pbill (T1 , T2 ). The higher the
value of the T-bill at maturity, the higher is the payoff from the forward contract. But
more it becomes more expensive to purchase T-bills for an investment between T1
and T2 . These two effects exactly cancel each other out, and the firm is guaranteed
the forward rate.
It is convenient for later reference to establish the result in the following:
Fact 5.8 Consider a forward contract in which one counterparty agrees to buy at a future
date T a zero coupon bond Pz (T, T ∗ ). The forward price is given by the forward discount
factor (multiplied by the notional).
Pzf w d (0, T, T ∗ ) = F (0, T, T ∗ ) × 100
5.3.1
(5.31)
A No Arbitrage Argument
It is useful to look at the arbitrage argument that makes Equation 5.31 true. Assume
Pzf w d (0, T, T ∗ ) > F (0, T, T ∗ ) × 100. Recalling that
F (0, T, T ∗ ) =
Pz (0, T ∗ )
Z(0, T ∗)
=
,
Z(0, T )
Pz (0, T )
an arbitrageur can:
1. Sell forward the zero coupon Pz (T, T ∗ ) at forward price Pzf w d (0, T, T ∗ ).
2. Short exactly F (0, T, T ∗ ) zero coupon bonds with maturity T for the amount
F (0, T, T ∗ ) × Pz (0, T ) = Pz (0, T ∗ ), the price of a zero coupon with maturity
T ∗.
3. Use the proceeds from step 2 to purchase one zero coupon with maturity T ∗ .
At time 0 there is no net cash flow for the arbitrageur. At time T the arbitrageur:
1. Delivers the zero coupon Pz (T, T ∗ ), which he or she owns, and receives Pzf w d (0, T, T ∗ ),
thereby closing the forward contract.
2. Covers the short position by paying F (0, T, T ∗ ) × 100.
The net cash flow at time T is then Pzf w d (0, T, T ∗ ) − F (0, T, T ∗ ) × 100 > 0. There is
no uncertainty or risk in this strategy, and thus it is an arbitrage. If Pzf w d (0, T, T ∗ ) <
F (0, T, T ∗ ) × 100, the reverse strategy also leads to an arbitrage, thus Equation 5.31 must
hold.
170
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
5.3.2 Forward Contracts on Treasury Bonds
Forward contracts can be written on any type of security. Of special interest are forward
contracts on Treasury notes and bonds.
Fact 5.9 Consider a forward contract in which one counterparty agrees to purchase at a
future date T a Treasury bond with coupon rate c and with maturity T ∗ > T . Let T1 , T2 ,
..., Tn = T ∗ be the coupon dates after T , the maturity of the forward contract. Then, the
forward price is given by
Pcf w d (0, T, T ∗ ) =
c × 100
c × 100
× F (0, T, T1 ) +
× F (0, T, T2 )
2 2
c × 100
+ 100 × F (0, T, Tn )
+... +
2
(5.32)
n
=
c × 100
×
F (0, T, Ti ) + 100 × F (0, T, Tn )
2
i= 1
=
c
×
P f w d (0, T, Ti ) + Pzf w d (0, T, Tn )
2 i= 1 z
(5.33)
n
(5.34)
Comparing the forward price Pcf w d (0, T, T ∗ ) in Equation 5.33 with the bond pricing
formula in Equation 2.13 in Chapter 2, the similarity is evident. Essentially, as discussed
in Definition 5.1 and shown in Equation 5.2, the forward discount factor F (0, T, Ti ) is
used in lieu of the current discount Z(0, Ti ) as the aim is to convert money at time Ti into
money at time T and not 0. Given our previous discussion about the forward discount
factor, Equation 5.33 should come to no surprise. However, it is beneficial to review a no
arbitrage argument that establishes Equation 5.33. In equation (5.33) suppose that
n
Pcf w d (0, T, T ∗ ) >
c
×
P f w d (0, T, Ti ) + Pzf w d (0, T, Tn )
2 i= 1 z
(5.35)
An arbitrageur can:
1. Sell the underlying bond forward at the forward price Pcf w d (0, T, T ∗ ).
n
2. Short N = 2c × i= 1 F (0, T, Ti ) + F (0, T, Tn ) zero coupon bonds maturing on
T.
3. Use the proceeds to purchase c/2 zero coupon bonds with maturities T1 to Tn −1 , and
1 + c/2 zero coupon bond with maturity Tn = T ∗ .
At time 0, there is no net cash flow for the arbitrageur. At time T , the arbitrageur must:
1. Sell all of zeros he holds and purchase the coupon bond Pc (T, T ∗ ). The law of one
price ensures that the price of this bond at T is the same as the value of sum of zeros.
The arbitrageur can then deliver the coupon bond in exchange of Pcf w d (0, T, T ∗ ).
2. Cover the short position in N zeros established in Step 2 by paying $100 × N .
The net cash flow is Pcf w d (0, T, T ∗ ) > $100 × N , which from Relation 5.35 is positive
yielding an arbitrage.
INTEREST RATE SWAPS
5.3.3
171
The Value of a Forward Contract
What is the value of a forward contract after initiation? Once again, the answer relies on the
cost (or profit) of closing the position. Just as an illustration, consider the forward contract
already discussed in Fact 5.9. We then have the following result:
Fact 5.10 Consider a forward contract in which one counterparty agrees to purchase at
a future date T a Treasury bond with coupon rate c and with maturity T ∗ > T . Let T1 ,
T2 , ..., Tn = T ∗ be the coupon dates after T . The value of the forward contract for every
t < T is
#
$
(5.36)
V f w d (t) = Z(t, T ) × Pcf w d (t, T, T ∗ ) − K
where K = Pcf w d (0, T, T ∗ ) is the delivery price agreed upon at initiation.
To see why Equation 5.36 holds, consider the investor who is long the forward contract, meaning that she committed to purchasing the coupon bond for a price K =
Pcf w d (0, T, T ∗ ). In order to close it before T , say at t < T , the investor can enter
short the contract to receive the same bond at the current forward price Pcf w d (t, T, T ∗ ).
Putting the two trades together, at T the investor will pay Pcf w d (0, T, T ∗ ) from the old
contract in exchange for the bond, which she will deliver to the counterparty of the new
t) contract for a price of Pc$f w d (t, T, T ∗ ). Overall, at T the investor will make
#(time
f wd
Pc (t, T, T ∗ ) − Pcf w d (0, T, T ∗ ) . This amount is known today (t), and thus its present
value represents the value of the forward contract, yielding (5.36).
5.3.3.1 The Forward Price and the Value of a Forward Contract It is easy to
get confused between the forward price and the value of a forward contract. The forward
price is not the price at which one trader can buy a forward contract. In fact, at initiation,
it does not cost anything to enter into a forward contract, as there is no money exchange
between the counterparties at time t = 0. The reason is that a forward contract is an
agreement today to exchange money in the future (and not today). The forward price is the
price at which we agree today to buy or sell the security in the future. Once the contract
has been initiated and the forward price has been set, though, then the value of the now old
forward contract changes over time with the variation of interest rates or the underlying
security. Thus, if a trader wants to enter into a forward contract at an old forward price
[e.g., Pcf w d (0, T, T ∗ ) in Equation 5.36], he may need to pay or receive money depending
on whether in the meantime current market conditions increased or decreased the forward
price.
5.4
INTEREST RATE SWAPS
Interest rate swaps have become the dominant interest rate over-the-counter (OTC) derivative security of modern financial markets. According to the Bank for International Settlements, as of December 2008 the interest rate swap market had a total market value of $8
trillion ($357 trillion notional), against $87 billion of forward rate agreements ($39 trillion
notional) and $1.1 trillion of OTC options ($62 trillion notional). These numbers may be
compared against the U.S. Treasury debt at that time, which was around $5.9 trillion. In
this section we look at the pricing of plain vanilla interest rate swaps, and the implications
for financial markets.
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INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Definition 5.6 A plain vanilla fixed-for-floating interest rate swap contract is an agreement between two counterparties in which one counterparty agrees to make n fixed payments
per year at an (annualized) rate c on a notional N up to a maturity date T , while at the
same time the other counterparty commits to make payments linked to a floating rate index
rn (t).6 Denote by T1 , T2 ,..., Tn = T the payment dates, with Ti = Ti−1 +Δ and Δ = 1/n,
the net payment between the two counterparties at each of these dates is
Net payment at Ti = N × Δ × [rn (Ti−1 ) − c]
(5.37)
The constant c is called swap rate.
The following example illustrates the cash flows involved in an interest rate swap with
semi-annual payments.
EXAMPLE 5.8
A firm and bank decide to enter into a fixed for floating, semi-annual, five-year swap
with swap rate c = 5.46% and notional amount N = $200 million. The reference
floating rate is the six months LIBOR. In this swap contract, the firm agrees to pay
to the bank every six months (Ti = 0.5, 1, 1.5, ..., 5) the amount
Cash flow from firm to bank at Ti = $200 m × 0.5 × 5.46% = $4.56 million
In exchange, the bank pays the firm at every Ti an amount that depends on the 6month LIBOR r2 (Ti−1 ). It is crucial to note that the reference rate for the payment
at time Ti is not the LIBOR at Ti , but the one determined six months before, at
Ti−1 = Ti − 0.5. This timing convention is important, as we shall see, to obtain
simple formulas to value swap contracts.
Cash flow from bank to firm at Ti = $200 m × 0.5 × r2 (Ti−1 )%
Table 5.4 illustrates the cash flows from the bank to the firm and vice versa. The
noteworthy point is that the cash flows from the bank to the firm in Column 3 vary
over time, and in particular they have a six months lag from the time the LIBOR, in
Column 2, is realized. In this particular instance, the firm would receive a negative
net cash flow, as the reference floating rate declined from 4.951% at initiation to a
much lower number.
It is important to reiterate that in a swap contract the two counterparties agree to exchange
cash flows in the future, not today. Therefore, there is a zero net cash flow at initiation of
the contract. As in forward contracts and in forward rate agreements, nobody buys or sells
a swap, in that no exchange of money occurs at initiation.
Before studying the fair pricing of swaps, it is useful to examine one more example in
which a firm uses a swap to hedge against interest rate risk.
that the subscript n on the interest rate rn (t) denotes the compounding frequency. For notational
convenience, I also drop the maturity index in the reference rate, that is, r n (t) = rn (t, t + Δ) when Δ = 1/n.
6 Recall
INTEREST RATE SWAPS
Table 5.4
173
Example Cash Flows in Fixed-for-Floating Swap
Time
LIBOR
Flow from Bank to Firm
Flow from Firm to Bank
Net Cash Flow to Firm
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
4.951 %
3.460 %
2.040 %
1.800 %
1.339 %
1.201 %
1.170 %
1.980 %
3.190 %
3.996 %
4.976 %
$ 4.951 m
$ 3.460 m
$ 2.040 m
$ 1.800 m
$ 1.339 m
$ 1.201 m
$ 1.170 m
$ 1.980 m
$ 3.190 m
$ 3.996 m
$ 5.460 m
$ 5.460 m
$ 5.460 m
$ 5.460 m
$ 5.460 m
$ 5.460 m
$ 5.460 m
$ 5.460 m
$ 5.460 m
$ 5.460 m
$ -0.509 m
$ -2.000 m
$ -3.420 m
$ -3.660 m
$ -4.121 m
$ -4.259 m
$ -4.290 m
$ -3.480 m
$ -2.270 m
$ -1.464 m
EXAMPLE 5.9
Today is t = 0 = March 1, 2001. Consider a firm that sold a piece of equipment
to a highly rated corporation, and it is then due to receive payments in 10 equal
installments of $5.5 million each over 5 years. The firm would like to use these $5.5
million semi-annual cash flows to hedge against the coupon payments the firm must
make to service a $200 million, floating rate bond that it issued some time in the
past, and also expiring in 5 years. Suppose that the floating rate on the corporate
bond is tied to the LIBOR, at LIBOR + 4 bps. The 6-month LIBOR on March 1,
2001 is currently at 4.95% and so the next interest rate payment the firm must make
is (4.95 + 0.04)%/2 × 200 million = $4.9 million. So, the next floating rate coupon
payment is covered. However, if the LIBOR were to increase by more than 0.51%
in the next 5 years, the cash flows from the installments would not be sufficient to
service the debt.
A solution is to enter into a fixed-for-floating swap with an investment bank, in
which the firm pays the fixed semi-annual swap rate c, over a notional of $200 million,
and the bank pays the 6-month LIBOR to the firm. On March 1, 2001, the swap rate
for a 5-year fixed-for-floating swap was quoted at c = 5.46%. So, in this case, the
net cash flow to the firm from the swap contract is
Net cash flow to the firm at Ti = $200 million ×
1
× [r2 (Ti−1 ) − 5.46%]
2
where r2 (t) is the six month LIBOR at time t.
Why does this swap resolve the problem?
Consider the net position of the firm (see Figure 5.3): at every Ti the firm
1. receives 5.5 million;
2. pays (r2 (Ti − 0.5) + 4bps)/2 × 200 million on its outstanding floating rate debt;
3. receives r2 (Ti − 0.5)/2 × 200 million from the bank as part of the swap; and
174
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Figure 5.3
Hedging with Swaps
time Ti
RECEIVABLE
⇓
⇓
⇓
⇓
$5.5 m
MARKET
$200
2
⇐= ⇐= ⇐=
FIRM
Floating rate bond
m × (r (T ) + 4bps)
2
i −1
Swap: fixed leg
m × 5.46%
$200
2
=⇒ =⇒ =⇒
⇐= ⇐= ⇐=
SWAP
DEALER
Swap: floating leg
m × r (T )
2
i −1
$200
2
Net Flow = 0
4. pays 5.46% × 0.5 × 200 million to the bank as part of the swap.
Summing up, the firm’s net cash flow position from the receivable, debt, and swap is
Total cash flow at Ti
=
5.5 million
(Receivable)
−(r2 (Ti − 0.5) + 4bps)/2 × 200 million
(Debt)
+0.5 × [r2 (Ti − 0.5) − 5.46%] × 200 million
(Swap)
=
5.5 − 0.04% × 100 − 5.46% × 100
=
0
That is, the firm is perfectly hedged: The risk in the fluctuations of the LIBOR
stemming from its liabilities has been eliminated by the swap (the firm receives the
LIBOR from the bank, and pays the LIBOR + .04% to bond holders). The remaining
fixed components sum up to zero. Figure 5.3 illustrates the flow of cash at every Ti .
The swap contract allows the firm to use a sequence of fixed cash flows (the receivables) to
hedge a sequence of floating cash flows (the LIBOR-based coupons). Using the terminology
developed in Chapter 3, the firm is facing a duration mismatch, as its assets (the receivables)
have long duration, while its liabilities (the floating rate bond) have low duration. The swap
contract eliminates the duration mismatch. Indeed, this flexibility to change the nature
of future cash flows without any payment at initiation has made the swap an excellent
instrument for corporations and governments to implement effective cash flow management
strategies. Section 5.5 further explores these applications.
INTEREST RATE SWAPS
5.4.1
175
The Value of a Swap
How do we value a swap? Consider the swap counterparty who makes the fixed payments
at (annualized) rate c every six months at dates T1 , T2 , ..., TM , and receives at the same
times floating rate payments linked to a floating rate r2 (Ti − 0.5). The sequence of net
cash flows for this counterparty is the same as the one for a portfolio that is long a floating
rate bond, and short a fixed rate bond with coupon c. The value of the swap is then readily
available. In fact, recall that in Chapter 2 we have already obtained the value of coupon
bonds (Section 2.4) and of floating rate bonds (Section 2.5). A direct application of those
results yields the value of the swap. More specifically, from the equivalence
Value of swap = Value of floating rate bond − Value of fixed rate bond
(5.38)
we obtain
V sw ap (t; c, T ) = PF R (t, T ) − Pc (t, T )
(5.39)
where V sw ap (t; c, T ) is the value of a swap at time t, with swap rate c and maturity T ,
PF R (t, T ) is the value of a floating rate bond as in Equation 2.40 in Chapter 2, and Pc (t, T )
is the value of fixed coupon bond, with coupon rate c, given in Equation 2.13 in the same
chapter.
At payment dates Ti , the value of the floating rate bond is PF R (Ti , T ) = 100. Also,
using the price of the bond Pc (Ti , T ) in Equation 2.13 we then obtain
⎛
⎞
M
c
V sw ap (Ti ; c, T ) = 100 − ⎝ × 100 ×
Z(Ti , Tj ) + Z(Ti , TM ) × 100⎠
2
j =i+ 1
(5.40)
EXAMPLE 5.10
Let’s revisit Example 5.9. The discount factors Z(0, T ) on March 1, 2001 are in the
second column of Table 5.5. We can apply formula in Equation 5.40, obtaining
V sw ap (0; c, T ) = 100 −
0.0546
× 100 × 8.69 + 0.7628 × 100 ≈ 0
2
(5.41)
The value of the swap in Example 5.9 is (almost) zero, reflecting the fact that it does
not cost anything to enter a swap contract at initiation.
5.4.2
The Swap Rate
How is the swap rate c determined? The contract specification implies that there is no
exchange of money at inception of the swap contract. This implies that at inception, the
value of the contract is zero. If the swap contract must have zero value at initiation of the
contract, Equation 5.39 immediately provides a rule to determine the value of the swap rate
c.
Fact 5.11 The swap rate c is given by that number that makes V sw ap (0; c, T ) in Equation
5.39 equal to zero. Rewriting the equation generically for any payment frequency n and
176
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Table 5.5 LIBOR Discounts and Swap Curve on March 1, 2001
Maturity T
Z(0, T )
Swap Curve
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.9758
0.9527
0.9289
0.9050
0.8808
0.8565
0.8327
0.8090
0.7858
0.7628
4.951
4.910
4.980
5.050
5.135
5.220
5.285
5.350
5.405
5.460
Data Source: Federal Reserve.
payment dates T1 , ... TM , we have
⎛
⎞
M
c
Z(0, Tj ) + Z(0, TM ) × 100⎠
V sw ap (0; c, T ) = 100 − ⎝ × 100 ×
n
j=1
(5.42)
Solving this equation for c we find
c=n×
1 − Z (0, TM )
M
j =1
Z (0, Tj )
(5.43)
As it can be seen, Equation 5.43 provides a relatively simple rule to compute the swap
rate, given a discount curve Z(0, Tj ) for the payment dates Tj .
EXAMPLE 5.11
In Example 5.10, given the discount factors in the second column of Table 5.5, the
swap rate 5.46% makes the swap value equal to zero. Thus, c = 5.46% is exactly the
proper swap rate, as it would be quoted by the bank in Example 5.9.
5.4.3
The Swap Curve
What are the appropriate discount factors Z (t, T ) to price swaps? Over the years, the swap
market grew so much that market forces determine the swap rate for every possible future
maturity. For instance, if corporations fear an increase in short-term rates and decide to
move to fixed rate financing, they may increase the demand for fixed-for-floating swaps, in
which they pay a fixed rate. This is equivalent to them selling fixed rate bonds, with the
implication that the price of fixed rate bonds decreases. That is, to keep the value of swaps
at par, the implicit coupon, the swap rates c, must increase. This increase in swap rates in
INTEREST RATE SWAPS
177
turn affects the time value of money, and thus the discount factors Z(t, Tj ) that are implicit
in swaps.
Definition 5.7 The swap curve at time t is the set of swap rates (at time t) for all maturities
T1 , T2 , ..., TM . We denote the swap curve at time t by c (t, Ti ) for i = 1, ..., M ;
Swap rates are quoted daily by swap dealers for swap contracts up to thirty years to
maturity.7 Given the size of the swap market, the swap curve c(t, T ) has become in fact
a reference point to determine the time value of money for financial institutions. Indeed,
given the set of swap rates c(t, Ti ), we can compute the implicit discount factors Z (t, Ti ),
by applying a bootstrap methodology, similar to the one discussed in Chapter 2, Section
2.4.2 for Treasury bonds, and in Section 5.1.3 for forward rates. Specifically, we can invert
Equation 5.43 and find that for i = 1
1
(5.44)
Z (t, T1 ) =
c(t,T 1 )
1+ n
while for i = 2, ..., M
Z (t, Ti )
1−
=
c(t,T i )
n
×
1+
i−1
j =1
c(t,T i )
n
Z (t, Tj )
(5.45)
EXAMPLE 5.12
The last column of Table 5.5 contains the swap curve data as of March 1, 2001. What
is the zero curve that is implicit in the swap curve? Starting with Equation 5.44, we
find
1
Z (0, 0.5) =
0.04951 = 0.9758
1+ 2
Moving to Equation 5.45 for i = 2, we have
Z (0, 1)
=
=
=
1−
0.0491
2
× Z (0, 0.5)
0.0491
2
1+
1 − 0.02455 × 0.9758
1 + 0.02455
0.9527
Similarly, for i = 3, we have
Z (0, 1.5)
0.0498
2
× (Z (0, 0.5) + Z (0, 1))
1 + 0.0498
2
1 − 0.0249 × (0.9758 + 0.9527)
=
1 + 0.0249
= 0.9289
=
1−
and so on. The result is the set of discount factors Z(0, T ) in the second column of
Table 5.5
7 Daily
data are available, for instance, at the Federal Reserve Board Web site: http://www.federalreserve.
gov/Releases/h15/data.htm .
178
5.4.4
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
The LIBOR Yield Curve and the Swap Spread
The previous section showed the computation of the discount factors Z(t, T ) that are
implicit in the market swap rates c(t, T ). In Chapter 2, Section 2.4.2, we obtained the
discount factors Z(t, T ) from Treasury bonds. It is a good question to wonder what is the
relation between the two. To differentiate between the two types of discount factors, let
us denote the discount factors obtained from swap rates as Z L (t, T ), where the superscript
“L” stands for “LIBOR”: It is customary to refer to the swap-rate implied discount factors
as the LIBOR discount, and its term structure as the LIBOR curve. The reason is that the
underlying floating rate is the LIBOR.
Panel A of Figure 5.4 shows the Treasury and LIBOR discount factors on January 4,
2005. Panel B plots the continuously compounded zero coupon yields that are implied
by the two discount curves in Panel A. Clearly, the LIBOR yield curve is higher than the
Treasury yield curve. One of the reasons for this difference is that the LIBOR curves also
contain a spread due to the probability of default of swap dealers. This spread is typically
very small, but it can become substantial during turbulent periods, such as the credit crisis
of 2007 - 2008. The case study at the end of this chapter further illustrates the variation in
the swap spread over time, with a particular emphasis to the events in 2007 - 2008.
Figure 5.4 Treasury and Swap Discount and Yield on January 4, 2005
A: Discount Functions
1
Treasury
Swap
0.95
Discount
0.9
0.85
0.8
0.75
0.7
0.65
0
1
2
3
4
5
6
Time to Maturity
7
8
9
10
B: Zero Coupon Yield Curves
5
Yield (%)
4.5
4
3.5
Treasury
3
2.5
Swap
0
1
2
3
4
5
6
Time to Maturity
7
8
9
10
Data Source: Federal Reserve and CRSP.
Panel A of Figure 5.5 plots the 5-year zero coupon yield implied by the swap curve and
from the Treasury curve, from 1986 to 2005. Panel B plots the time series of the 5 year
swap spread, that is, the difference between the two zero coupon yields. As we can see
from both panels, the swap spread varies substantially over time. This variation prompted
proprietory trading desks and hedge funds to speculate on the fact that the swap spread is
INTEREST RATE SWAPS
179
unlikely to become negative, and unlikely to move to infinity. When the swap spread is
large, for instance, speculators may bet on the fact that the spread will shrink in the future
by taking a short position on the Treasury bonds, and enter in a fixed-for-floating swap, as
a fixed rate receiver. We study a swap spread trade in the case analysis at the end of this
chapter (see Section 5.8).
Figure 5.5 The 5-year Zero Coupon Treasury and Swap Yield
5−year Treasury and Swap Zero Yield
Treasury
Swap
Yield (%)
10
8
6
4
2
1986
1988
1990
1992
1994
1996
1998
2000
2002
2004
2006
2000
2002
2004
2006
5−year Swap − Treasury Spread
1.2
Spread (%)
1
0.8
0.6
0.4
0.2
0
1986
1988
1990
1992
1994
1996
1998
Data Source: Bloomberg and CRSP.
5.4.5
The Forward Swap Contract and the Forward Swap Rate
The same way it is possible to lock in a future interest rate today by entering into a forward
rate agreement, it is also possible to lock in a future swap rate by entering into a forward
swap contract.
Definition 5.8 The forward swap contract is a contract in which two counterparties agree
to enter into a swap contract at a predetermined future date and for a predetermined swap
rate f s , called the forward swap rate.
The next example illustrates the forward swap contract.
EXAMPLE 5.13
Consider Example 5.9, but assume now that on March 1, 2001, the firm signed a
contract to deliver one year later (on March 1, 2002) a large piece of equipment.
The payment will be made in 8 equal installments of $5.5 million each over the
next 4 years, starting on September 1, 2003. Assume the firm plans to use these
180
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
cash inflows to meet the payments of a floating rate bond issued some time in the
past. As explained in Example 5.9, the firm can enter into a fixed-for-floating swap
in which it pays fixed and receives floating. The problem is that the firm will start
receiving payments much further in the future, and therefore it will need to enter into
such a fixed-for-floating swap one year from now, on March 1, 2002. The firm is
worried however that the 4-year swap rate may increase between now and March 1,
2002, an event that may unduly increase its cash outflows from the hedging program.
Therefore, the firm decides to enter into a forward contract with a bank, in which
the bank and the firm agree today that the 4-year swap rate in the future will be
f2s = 5.616%, to be paid semi-annually in exchange of the 6-month LIBOR.8
The question then is the usual: How can the bank commit today to enter into a swap contract
in the future at a given swap rate f2s ? To answer this question, we may recall that the value
of a fixed-for-floating swap in which a counterparty receives the fixed rate c can be regarded
as a portfolio that is long a coupon bond, with coupon rate c, and short a floating rate bond,
that has value of 100 at reset dates. The payoff from entering a forward swap contract is
then the same as entering into a forward contract to purchase a fixed rate bond with coupon
rate c for par value 100. That is, the payoff from a forward swap is
Payoff forward swap = Pc (T, T ∗ ) − 100
where
(5.46)
m
Pc (T, T ∗ ) =
c × 100
×
Z(T, Tj ) + 100 × Z(T, T ∗ )
2
j=1
and T1 , T2 , ..., Tm are the swap’s reset dates, with Tm = T ∗ .
What is the value today of the payoff in Equation 5.46?
Notice that this payoff is the same as the payoff of a forward contract to receive a coupon
bond at T for a delivery price K = 100. We discussed the valuation of such a forward
contract in Section 5.3.3. Specifically, the value of such forward contract is provided in
Equation 5.36 in which the delivery price happens to be equal to par, K = 100. The value
of the forward swap contract today is then
$
#
(5.47)
V (0, T ) = Z(0, T ) Pcf w d (0, T, T ∗ ) − 100
where from Equation 5.33
m
c × 100
×
F (0, T, T ∗ ) + 100 × F (0, T, Tm )
2
j=1
Pcf w d (0, T, T ∗ ) =
(5.48)
While in a standard forward contract the delivery price is chosen to make the value of
the forward contract equal to zero at initiation, in a forward swap contract it is the swap
rate c that is chosen to make the value of the forward contract equal to zero. Thus, we must
look for c such that V (0, T ) = 0 in Equation 5.47. This condition implies
m
c × 100
×
F (0, T, Tj ) + 100 × F (0, T, T ∗ ) = 100
2
j=1
8 As
usual, the subscript “2” denotes the semi-annual compounding frequency.
(5.49)
INTEREST RATE SWAPS
181
Solving for c, and denoting the result as the forward swap rate f2s (0, T, T ∗ ), we obtain
f2s (0, T, T ∗ )
=
2×
1 − F (0, T, T ∗ )
m
j =1 F (0, T, Tj )
(5.50)
Comparing Equation 5.50 with Equation 5.43 (for n = 2), we see that the forward swap
rate is the swap rate that is implicit in the forward curve. That is, in the same way we obtain
the current swap rate from the discount curve Z(0, Tj ), we can compute the the forward
swap rate from the forward discount curve F (0, T, Tj ). We summarize this result next for
future reference:
Fact 5.12 The forward swap rate of a forward swap contract to enter into a swap at
time T , with maturity T ∗ , payment frequency n, and payment dates T1 , T2 ,...,Tm (with
Tm = T ∗ ) is given by
fns (0, T, T ∗ ) = n ×
1 − F (0, T, T )
m
j =1 F (0, T, Tj )
(5.51)
EXAMPLE 5.14
Returning to Example 5.13, we can use the LIBOR discount data in Table 5.5,
reported also in the Column 2 of Table 5.6, to compute the forward swap rate to enter
on March 1, 2002 (one year from now) into a 4-year swap. For every Tj = 1.5, 2, ...,
5, we can compute the forward discount factor
F (0, 1, Tj ) =
Z(0, Tj )
Z(0, 1)
This calculation is in Column 3 of Table 5.6. Applying the formula in Equation 5.50
we obtain
f2s (0, 1, 5) = 5.616%
5.4.6
Payment Frequency and Day Count Conventions
In the previous sections we considered the case in which the two counterparties make
payments at the same time, for instance, every six months. Many swap contracts, however,
have payment dates at different frequency. For instance, a commonly used fixed-for-floating
contract specifies that the floating rate is tied to the 3-month LIBOR and pays at a quarterly
frequency, while the fixed rate counterparty pays at a semi-annual frequency. Different
frequencies in payments pose no particular problems in the valuation of swaps, because the
value of a swap is given by the difference in value between the floating leg and the fixed
leg, and we are able to value them independently. The case study in Section 5.8 at the end
of this chapter further discusses this issue.
One more issue to explore concerns the day count conventions used in the computation
of cash flows. In the previous section we implicitly assumed an actual/actual convention,
for simplicity. To be precise, the floating leg uses the actual/360 day count convention,
182
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Table 5.6 LIBOR Discounts and Forward Discounts on March 1, 2001
Maturity T
Z(0, T )
F (0, 1, T )
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0.9758
0.9527
0.9289
0.9050
0.8808
0.8565
0.8327
0.8090
0.7858
0.7628
0.9750
0.9499
0.9245
0.8990
0.8740
0.8492
0.8248
0.8007
Data Source: Federal Reserve.
while the fixed leg uses the 30/360 day count convention. Therefore, a little adjustment
to the payoffs has to be made. Although day count conventions are important in order
to compute the exact values of fixed income securities and for day-to-day trading, we
will often abstract from these institutional details in our calculations. The reason is that
they distract the attention from more fundamental concepts in fixed income, such as no
arbitrage and the relative values of fixed income instruments. The understanding of the risk
and return of fixed income instruments and the forces that keep their prices close to each
other is far more important than delving into the day count conventions, or other minor
institutional details, which computers compute automatically anyway. To put it differently,
it is unlikely that financial institutions may be at risk of losing billion of dollars because of
the misunderstanding of day count conventions, but it is likely they may face these risks if
they do not understand the rules of arbitrage and the fundamental variations of fixed income
instruments.
5.5 INTEREST RATE RISK MANAGEMENT USING DERIVATIVE
SECURITIES
Derivatives such as interest rate forwards and swaps are particularly useful in performing an
effective interest rate risk management. We already illustrated the use of these securities in
risk management in Examples 5.1 to 5.9. In this section we extend the concepts discussed
in Chapter 3 to illustrate some of the uses of derivatives for asset-liability management or
duration matching. Indeed, we now show that by carefully choosing the characteristics of
the derivative security, the duration of a portfolio that contains the derivative security can
be tailored with a great deal of accuracy to the needs of a corporation.
Section 3.4 in Chapter 3 introduces the issue of asset-liability duration mismatch. We
can use derivatives, such as forwards and swaps, to effectively perform duration matching
strategies. In fact, Example 5.9 illustrates exactly such a duration mismatch: The firm in
that example is paying a floating rate on its bonds, but it is receiving a fixed coupon from
the receivables. That is, it effectively has a duration mismatch, featuring assets with long
duration and liabilities with short duration. As the example illustrates, a carefully crafted
183
INTEREST RATE RISK MANAGEMENT USING DERIVATIVE SECURITIES
fixed-for-floating swap is able to completely eliminate the duration mismatch, as the firm’s
total net cash flows are exactly zero.
As discussed in Section 3.4 in Chapter 3, corporations and financial institutions have
very complex assets and liabilities. In this case, an immunization strategy is better suited to
deal with a potential duration mismatch. In particular, a financial institution may then use
derivative securities to close the duration gap. For instance, a fixed-for-floating swap does
not cost anything to enter into, but it can dramatically change the duration of a portfolio.
More specifically, since a swap can be considered a long-short portfolio (long a floating
rate bond and short a fixed rate bond), as discussed in Chapter 3, Example 3.6, the dollar
duration of the swap can be computed as
$
$
$
Dsw
ap = Df loatin g − Df ixed
where Df$ ixed and Df$ loatin g are the dollar durations of the fixed and floating bonds that are
underlying the swap. Denoting N the notional on the swap, we may choose the notional
amount so that the duration of equity is zero, that is, such that a parallel shift in the term
structure has no impact on the equity value. This objective is accomplished if the following
equation holds:
$
$
DE$ = DA$ + N Dsw
ap − DL = 0
(5.52)
We now illustrate the use of swaps for asset-liability management in the next example.
EXAMPLE 5.15
Consider Example 3.11 in Chapter 3. That example, recall, involves a hypothetical
financial institution with a total asset size of around $2.4 billion and assets’ dollar
duration of $19.74 billion. The firm has also total liabilities at $1.8 billion with a
liabilities’ dollar duration of only $5 billion. The market value of equity is $600
million, but with a dollar duration is $14.740 billion. The implications of this
mismatch between the assets and liabilities dollar durations is that a parallel upward
shift in interest rates of 1% generates a decline in assets far greater than in liabilities,
implying an equity decline of $147.4 million. In percentage, this corresponds to a
24% decline in market value of equity.
How can derivatives, and in particular swaps, help stabilize the value of equity?
Assume that the term structure of interest rates is flat at 4% (semi-annually compounded). In this case, the current swap rate is also 4%. By using the same steps as in
Example 3.6 in Chapter 3, the dollar duration of a fixed-for-floating swap (receiving
fixed) can be found to be $784 for $100 of notional.9 Clearly, the financial institution
has a duration of assets that is higher than the one of its liabilities, and therefore
would like to enter into a swap in which it pays fixed and receives floating, which
has the opposite duration of −$784 (per $100 notional). From Equation 5.52, we can
choose the notional N to make the dollar duration of equity DE$ = 0. The notional
that solves the equation is $1.889 billion.
9 The
long-short portfolio strategy in Example 3.6 in Chapter 3 exactly represents a fixed-for-floating swap.
184
5.6
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
SUMMARY
In this chapter, we covered the following topics:
1. Forward discount factors: This is the discount factor implicit in the current yield
curve to discount dollars in the far future to the nearer future, but not today. It
provides the “exchange rate” between dollars at T2 for dollars at T1 < T2 .
2. Forward rate: The forward rate is an interest rate implicit in the current term structure
of interest rates determining the fair rate of interest for an investment (or a loan) at
a future date T1 with payment at a later date T2 . It is determined by the forward
discount factor.
3. Forward rate agreement (FRA): A FRA is a contract between two counterparties
to exchange one cash flow in the future, namely, the forward rate in exchange of
the future spot rate. Forward rate agreements are widely used to hedge againsts
variations in interest rates.
4. Forward contract: This is a contract between two counterparties in which they agree
that at some predetermined date, they will exchange a security, such as a Treasury
note, for a cash price that is also predetermined at initiation of the contract. Forward
contracts are equivalent to forward rate agrements.
5. Swap: A swap is a contract between two counterparties to exchange cash flows in
the future. In a fixed-for-floating swap a counterparty pays a fixed coupon while the
other pays a rate linked to a floating rate, typically the LIBOR rate. The fixed rate is
called the swap rate, and it is set at initiation of the contract so that the value of the
swap is zero.
6. Swap curve: The relation between swap rates and the maturity of the underlying
swap is called the swap curve.
7. LIBOR curve: The discount curve implicit in LIBOR based instruments, such as
swaps, differs from the Treasury curve as it embeds the risk of default of swap
counterparties.
8. Forward swap contract: A contract between two counterparties in which they agree
to enter into a given swap contract in the future, with a predetermined swap rate,
called the forward swap rate, and has a predetermined maturity.
5.7 EXERCISES
1. On May 15, 2000 the term structure of interest rates is as shown in Table 5.7.
Compute the discount factors Z(0, T ), the forward discount factors F (0, T − Δ, T ),
and the forward rates f (0, T − Δ, T ), where Δ = 0.25.
2. Table 5.8 contains the continuously compounded forward rates f (0, T − Δ, T ),
where Δ = 0.25. The first entry is the current spot rate, as for T = 0.25 we have
f (0, 0, 0.25) = r(0, 0.25). Compute the forward discount factors F (0, T − Δ, T ),
the current discount factors Z(0, T ), and the current term structure of interest rates.
EXERCISES
Table 5.7
185
The Term Structure of Interest Rates on May 15, 2000
Maturity
Yield (c.c.)
Maturity
Yield (c.c.)
Maturity
Yield (c.c.)
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
6.17%
6.52%
6.32%
6.71%
6.76%
6.79%
6.77%
6.72%
6.72%
6.79%
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
6.78%
6.76%
6.77%
6.76%
6.63%
6.77%
6.77%
6.71%
6.66%
6.70%
5.25
5.50
5.75
6.00
6.25
6.50
6.75
7.00
6.71%
6.63%
6.69%
6.62%
6.63%
6.61%
6.58%
6.57%
Notes: Yields calculated based on data from CRSP.
Table 5.8
A Term Structure of Forward Rates
Maturity
Forward Rate
Maturity
Forward Rate
Maturity
Forward Rate
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
3.53%
3.58%
4.19%
3.99%
4.54%
5.00%
4.76%
5.88%
5.30%
4.92%
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
6.09%
5.29%
6.48%
6.20%
6.34%
6.00%
5.99%
6.58%
6.26%
6.69%
5.25
5.50
5.75
6.00
6.25
6.50
6.75
7.00
6.12%
5.70%
6.81%
6.50%
6.59%
7.06%
6.87%
6.37%
186
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Table 5.9
Two Discount Curves
August 15, 2000
Maturity
Z(0, T )
0.25
0.50
0.75
1.00
0.9844
0.9690
0.9531
0.9386
November 15, 2000
Maturity
Z(0, T )
0.25
0.50
0.75
1.00
0.9848
0.9692
0.9545
0.9402
Source: Bloomberg.
3. Today is May 15, 2000 and the continuously compounded term structure of interest
rate is shown in Table 5.7. You are faced with the two investment strategies below.
Is there an arbitrage opportunity?
• Invest $100 million in 2.5-year zero coupon bonds.
• Invest $100 million in a 1-year zero coupon bond and agree with the bank to
invest the proceeds for the following 1.5 years at the quoted forward rate, which
is: f2 (0, 1, 2.5) = 7.56%.
4. On May 15, 2000 you enter into a 1-year forward rate agreement (FRA) with a bank
for the period starting November 15, 2000 to May 15, 2001. You know that currently
the price of the 6-month zero coupon is $96.79 and the price of the 1-year zero
coupon is $93.51.
(a) What is the agreed-upon forward rate in the transaction?
(b) What is the value of the forward at inception?
5. Consider Exercise 4 again. Three months later (August 15, 2000) you have second
thoughts and consider that maybe you should get out of the transaction. You receive
the data in the first two columns of Table 5.9.
(a) What is the value of the FRA on August 15, 2000?
(b) Consider now November 15, 2000.
i. What is the value of the FRA now?
ii. What is the 6-month semi-annual rate?
iii. What will the balance to be paid be at the end of the FRA?
6. On May 15, 2000, a company is interested in purchasing $50 million worth of 1
1/2-year zero coupon Treasuries with the proceeds of a sale of equipment to take
place in 6 months. The company is interested in locking in the price of the Treasuries
today through a forward contract. Use the data in Table 5.10 to answer the following:
• What would the forward price be of the Treasuries?
• How many bonds will the company purchase?
EXERCISES
Table 5.10
Two Discount Curves
May 15, 2000
Maturity
Z(t, T )
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
0.9847
0.9679
0.9537
0.9351
0.9189
0.9031
0.8882
0.8742
187
November 15, 2000
Maturity
Z(t, T )
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
0.9848
0.9692
0.9545
0.9402
0.9269
0.9147
0.9023
0.8897
Data Source: CRSP.
7. Consider Exercise 6. Six months have now passed so that today is November 15,
2000. You want to calculate the payoff from being long the forward contract. Using
the data in Table 5.10 answer the following
(a) What is the amount of the payoff if you are long the forward contract?
(b) Do you make money or lose money?
8. Consider the following transaction:
(a) On June 30, 2008 a financial institution purchases $100 million worth of a
2-year Treasury note paying a 2.88% coupon priced at $100.50. At the same
time the financial institution enters into a forward contract with the seller of
the Treasury note in which it will sell the same T-note three months later (no
accrued interest payments will be made). Use the (continuously compounded)
yield curve on June 30, 2008 in Table 5.11 to answer the following:
i. How many bonds does the financial institution purchase?
ii. What is the quoted price for the Treasuries in three months?
(b) Recall the definition of a repo agreement (Chapter 1). In this transaction the
buyer agrees to acquire a security and to resell it to the seller at a specified time.
This is similar to the outright purchase of a Treasury plus the forward contract.
i. Calculate the implied repo rate for this security.
ii. The actual 3-month repo rate for June 30, 2008 was 2.05%. Is this the
same as the implied repo rate you calculated? Explain.
iii. Assuming that this is a pure arbitrage opportunity, what steps would the
financial institution have to follow to take advantage of it?
iv. Is the implied repo rate the same as the return on investing in a 3-month
bond?
(c) Consider the following: The repo agreement simply uses the security as collateral and does not depend on the maturity of the bond, yet the transaction we
described depends on the maturity, since it affects both the spot price on the
bond and the forward price. Additionally, there are coupon payments in this
188
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Table 5.11
The Yield Curve on June 30, 2008
Maturity
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Yield (c.c.)
1.71%
2.09%
2.29%
2.37%
2.32%
2.38%
2.48%
2.61%
Source: Bloomberg.
Table 5.12
The Yield Curve on May 5, 2008
Maturity
0.25
0.5
0.75
1
1.25
1.5
1.75
2
Yield (c.c.)
2.70%
2.76%
2.86%
2.95%
3.09%
2.98%
3.07%
3.20%
Source: Bloomberg.
bond which also affect the value of the spot price and the forward price. It
seems paradoxical that while the repo rate depends only on the length of the
loan, the equivalent transaction takes into account maturity of the collateral
(Treasury) and its coupon payments.
i. Compute the implied 3-month repo rate using a 2-year zero coupon bond
(obtain the spot and forward prices from the discounts you obtain from the
yields).
ii. Compute the implied 3-month repo rate using all zero coupon securities
with maturities from 3 months up to 1-3/4 years.
iii. Does the implied repo rate change? Why or why not?
9. Today is May 5, 2008, and the (continuously compounded) yield curve is given in
Table 5.12. Calculate the semi-annual swap rate for all maturities between 6 months
and 2 years (every six months).
10. Today is January 2, 2008. The LIBOR curve is shown in the first column of Table 5.13.
You decide to enter into a 1-year fixed-for-floating swap agreement with quarterly
payments and $100 million notional.
(a) What is the 1-year swap rate for a quarterly fixed-for-floating swap?
(b) What is the value of the agreement at inception?
(c) Calculate the value of the swap for each one of the subsequent dates in Table
5.13.
11. As of December 2, 2008, the 30-year swap spread had been negative for a whole
month. In particular, on that day, the 3-month repo rate was 0.5%, the LIBOR rate
was 2.21%, the 30-year swap rate was 2.85%, and the semi-annually compounded
yield-to-maturity of the 4.5% Treasury bond maturing on May 15, 2038 was 3.18%.
(a) Is there an arbitrage? Discuss the swap spread trade that you would set up to
take advantage of these rates.
CASE STUDY: PIVE CAPITAL SWAP SPREAD TRADES
Table 5.13
189
The LIBOR Curve: January, 2008 – October, 2008
Months
2-Jan
1-Feb
3-Mar
1-Apr
1-May
2-Jun
1-Jul
1-Aug
1-Sep
1-Oct
1
2
3
4
5
6
7
8
9
10
11
12
4.57%
4.64%
4.68%
4.65%
4.61%
4.57%
4.50%
4.42%
4.35%
4.29%
4.24%
4.19%
3.14%
3.11%
3.10%
3.07%
3.05%
3.02%
2.97%
2.92%
2.88%
2.85%
2.83%
2.82%
3.09%
3.04%
3.01%
2.97%
2.91%
2.86%
2.81%
2.76%
2.71%
2.68%
2.65%
2.63%
2.70%
2.69%
2.68%
2.66%
2.64%
2.62%
2.58%
2.54%
2.51%
2.50%
2.48%
2.47%
2.72%
2.76%
2.78%
2.82%
2.85%
2.88%
2.90%
2.92%
2.93%
2.95%
2.97%
2.98%
2.46%
2.57%
2.68%
2.75%
2.83%
2.90%
2.94%
2.98%
3.02%
3.06%
3.10%
3.14%
2.46%
2.65%
2.79%
2.89%
3.01%
3.12%
3.15%
3.19%
3.22%
3.25%
3.29%
3.32%
2.46%
2.66%
2.79%
2.89%
3.00%
3.08%
3.10%
3.12%
3.14%
3.17%
3.20%
3.22%
2.49%
2.68%
2.81%
2.94%
3.02%
3.11%
3.12%
3.14%
3.15%
3.16%
3.18%
3.20%
4.00%
4.05%
4.15%
4.09%
4.07%
4.04%
4.04%
4.04%
4.04%
4.04%
4.04%
4.04%
Source: Bloomberg.
(b) Assuming that the U.S. government is less likely to default than swap dealers,
how can you rationalize these rates? What risks would setting up this trade
involve? Discuss. (Recall that there was an ongoing credit crisis).
5.8
CASE STUDY: PIVE CAPITAL SWAP SPREAD TRADES
On June 30, 2006 PiVe Capital10 û- a small hedge fund û- was looking to make a profit on the
swap spread. The swap spread is the difference in coupon from a swap and a Treasury, and
it is present because of the higher probability of default of a swap counterparty than that of
the government. Although it varies significantly over time, as Figure 5.5 shows, whenever
it is positive it implies that receiving fixed from the swap and shorting the Treasury would
generate a positive cash flow over time. Clearly, this is only one side (the fixed side). The
second part has to do with floating rates. To receive fixed coupons in the swap, the fund
has to pay LIBOR over time, which must be taken into account in the trade. However,
partly offsetting this cash outflow, PiVe Capital can short the Treasury note through the
repo market in a reverse repo transaction,11 thereby obtaining the repo rate from a repo
dealer as an inflow of cash.
More specifically, on June 30, 2006 the market presented the following data:12
• 3-month LIBOR: 5.5081%.
• 3-month repo rate: 5.27%.
• 5-year swap rate: 5.69%.
• 5-year T-note with a 5.125% coupon priced at $100.1172.
10 This
is a fictituous name. Any reference to existing companies is purely coincidental.
Chapter 1 and the Orange County case in Chapter 3.
12 LIBOR, Repo and Swap rate data are from Bloomberg. Treasury securities data are excerpted from CRSP
c 2009 Center for Research in Security Prices (CRSP), The University of Chicago Booth
(Daily Treasuries) School of Business.
11 See
190
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Table 5.14
3-month LIBOR - Repo Spread Distribution: January, 2000 - June, 20006
Mean
St.Dev.
1%
5%
10%
25%
0.21%
.07%
0.05%
0.13%
0.14%
0.17%
Percentiles
50%
0.20%
75%
90%
95%
99%
0.25%
0.29%
0.31%
0.37%
Data Source: Bloomberg.
The swap spread is typically computed by comparing the yield to maturity of the Treasury
note with the swap rate of the same maturity. The yield to maturity of the 5-year T-note
can be computed using the formula in Equation 2.30 in Chapter 2, that is, by solving for y
in the formula
⎤
⎡
10
1
1
c
⎦
+
100.1172 = 100 × ⎣ ×
2 j = 1 (1 + y/2)j
(1 + y/2)10
We obtain y = 5.10%, implying that the spread is
SS = 5.69% − 5.10% = 0.59%
or 59 basis points. That is, receiving the fixed rate from the swap and paying the yield in
short T-notes would return 59 basis points per year.
To secure this return of 59 basis points, PiVe Capital has to pay the LIBOR rate, as part
of the swap, and receives the repo rate, as part of the reverse repo transaction. The spread
between LIBOR and repo (LRS) is
LRS = 5.5081% − 5.27% = 0.2381%
The net spread is SS − LRS = 35.19 bps. This spread is not as large as it had been in
the past, but the LIBOR-repo spread has been historically relatively stable at around 21
bps and so this net spread (SS − LRS) still appears a relatively safe trade. Indeed, using
daily data from January 2000 to June 2006, the distribution of the LIBOR-repo spread is
shown in Table 5.14. According to historical data, then, the mean LRS was 21 bps, and the
median (50% percentile) was 20 bps. In addition, the 95 and 99 percentiles were still only
31 bps and 37 bps, respectively. That is, there is only 1% probability that the LRS would
be above 37 basis points. Even in these extreme circumstances, the net spread would be
at least 0.59% − 0.37% = 22 bps, still a relatively good spread. Moreover, there are solid
chances that the LRS would decline, as its 10% percentile is only 14 bps.
PiVe Capital finally decided that, given historical trends, the net spread was high enough.
To recap, in order to reap these benefits PiVe must enter into the following transactions:
1. Short the 5-year bond through a reverse repo transaction. In cash flow terms, the
fund would receive the repo rate and pay the coupon.
2. Enter into a fixed-for-floating swap, in which PiVe would receive fixed and pay
LIBOR.
CASE STUDY: PIVE CAPITAL SWAP SPREAD TRADES
5.8.1
191
Setting Up the Trade
5.8.1.1 Reverse Repo PiVe Capital wants to set up a $100 million trade. Because
the T-note trades at $100.1172, it will need to sell
N=
$100million
= 998, 829 Treasury notes (for $100 par value)
100.1172
(5.53)
Assuming zero haircut for simplicity, PiVe Capital then borrows N = 998, 829 5-year
T-notes from the repo dealer, sells them in the cash market for $100 million, and gives
this cash to the repo dealer. Assume PiVe enters into a term reverse repo with 3-months
maturity. After three months, then, the repo dealer will have to pay PiVe Capital $100
million × Repo rate, where the repo rate = 5.27%, as above. At this time, if PiVe Capital
wants to keep the short position in the T-note, it must roll over the reverse repo position with
the repo dealer. Since in the meantime the price of the T-note will have changed, likely the
amount of cash with the repo dealer would have changed as well. Assume for simplicity
that the two counterparties simply roll over the $100 million loan for an additional three
months, irrespective of the price of the T-note. This simplifying assumption allows us to
better compare the reverse repo transaction to the fixed-for-floating swap, described below.
Because the repo dealer keeps the $100 million at every reset date (i.e., every three
months), the total cash flow to PiVe Capital every quarter is
Reverse repo CF(t) =
$100 mm
4
$100 mm
4
× r(t − 0.25) − N × 100 ×
× r(t − 0.25)
5.125%
2
if t is a coupon
date otherwise
where N is the number of 5-year T-notes, 5.125% is the T-note coupon rate, and r(t) denotes
the 3-month repo rate. Note that to keep things simple, we approximate the quarter by
“1/4” for the computation of repo cash flows, and the semester by “1/2” for the computation
of the T-note cash flows. In reality, the day count convention for repo rates is Actual/360,
and the day count convention for T-notes is Actual/Actual, and so some adjustments to the
cash flows would be necessary.
5.8.1.2 Fixed-for-Floating Swap PiVe Capital has to enter into a 5-year, fixed-forfloating swap in which it pays the 3-month LIBOR rate and receives the 5-year fixed swap
rate. In a plain vanilla fixed-for-floating swap, the floating payments occur at quarterly
frequency, while the fixed payments occur at semi-annual frequency, therefore exactly
matching the payment frequency of the reverse repo transaction discussed earlier. The
swap cash flows are then given by
Swap CF(t) =
$100 mm ×
5.69%
2
− $1004mm × (t − 0.25)
− $1004mm × (t − 0.25)
if t is fixed payment date
otherwise
where (t) is the 3-month LIBOR rate at time t. Again we approximate the quarter by
“1/4,” instead of Actual/360 for LIBOR. The fixed component of swaps is on 30/360 basis,
and so “1/2” is the correct value in this case.
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INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
5.8.2 The Quarterly Cash Flow
What is the quarterly cash flow of the trade? It is useful to decompose the quarterly cash
flow in two components, the swap spread (SS) component and the LIBOR-repo spread
(LRS) component. We know that every six months, the swap spread component of the cash
flow is
SS CF every six months
=
=
5.125%
5.69%
− N × 100 ×
2
2
$285, 499.73
$100 mm ×
where N is the number of T-notes that PiVe Capital shorted. This cash flow is fixed and
accrues every six months. It does not depend on extant interest rates, and it will keep
flowing to PiVe Capital until maturity.
The second component is the LIBOR-repo spread, which every quarter is given by
LRS CF every three months =
$100 mm
× (r(t − 0.25) − (t − 0.25))
4
where we recall that r(t) denotes the repo rate at t and (t) the LIBOR rate at t. As
mentioned, on June 30 the r(0) = 5.27% while (0) = 5.5081%, implying an initial cash
flow In September 30, 2006 equal to
LRS CF on September 30, 2006 = −$59, 525.00
(5.54)
Since the LIBOR-repo spread is relatively stable, PiVe Capital expected to receive approximately
Total expected net cash flow per year
=
2 × $230, 499.43 − 4 × $59, 525.00
=
$699, 098.85
Indeed, for the first year, things seem to play out well. Figure 5.6 shows the net cash
flows per quarter (the vertical bars) as well as the cumulative cash flows from the start of
the trade (the solid line).
The first shock however was in December 2007, when the total net cash flow, albeit
positive, was much smaller than before. On March 2008 the net cash flow, which was only
given by the LIBOR-repo spread component, was substantially negative, eating up most
of the cumulative cash flows made so far by PiVe Capital. On June 30, 2008 matters did
not improve, as the total inflow from the swap spread coupons was almost entirely paid in
the LIBOR-repo spread. In September 30, 2008 the last cash flow was extremely negative,
dragging to negative the entire cumulative sums.
What happened? Figure 5.7 plots the daily swap spread13 and LIBOR-repo spread from
July, 2000 to June, 2008 at the daily frequency. The first dashed vertical line corresponds
to June 30, 2006, the beginning of the trade. As mentioned, the LIBOR-repo spread had
been relatively stable before June, 2006, and (almost) always below 50 bps for the entire
sample. The swap spread on June 30, 2006 was at the peak of recent years, although still
lower than its peak in 2000. Recall though that the movement of the swap spread itself is
13 The
swap spread in this figure is computed as the difference between the 5-year constant maturity rate from the
Federal Reserve Board and the 5-year swap rate. Data are from the Federal Reserve.
CASE STUDY: PIVE CAPITAL SWAP SPREAD TRADES
Figure 5.6
193
Net Cash Flows from Swap Spread Trade
500
Quarterly Cash Flow
Cumulative Sum
400
Cash Flow (thousand USD)
300
200
100
0
−100
−200
−300
09/30/06 12/31/06 03/31/07 06/30/07 09/30/07 12/31/07 03/31/08 06/30/08 09/30/08
not affecting the cash flows after the initiation of the trade, because both the swap rate and
the coupon are then fixed. The horizontal line in the figure indeed denotes the 59 basis
points that PiVe Capital would make annually from the difference between the fixed swap
rate and the fixed coupon.
The problem was that the credit crisis that started in August of 2007 unexpectedly pushed
up the LIBOR-repo spread to an unprecedented 1.2%. Because borrowing and lending in
the LIBOR market is on an uncollateralized basis, fear of counterparty default and the
ensuing credit crunch made the LIBOR rate diverge substantially from the repo rate. The
latter is in fact perceived as a “safer” rate, as it is collateralized. Unfortunately, PiVe Capital
had to pay LIBOR and receive repo, and the large spread generated a significant outflow
of money over time. Note that the LIBOR rate in fact decreased in August and September,
as a result of the Federal Reserve cut of its reference interest rates. Yet, the spread is what
matters for PiVe, and not the level of the LIBOR per se.
5.8.3
Unwinding the Position?
Given the large quarterly payments due to the increase in the LIBOR-repo spread, the
principals of PiVe Capital started thinking about whether they should unwind their position.
The first question, of course, would be whether such an unwind of the reverse repo and the
swap would generate a net inflow or outflow of cash. We now compute the value of the
reverse repo and of the swap to calculate the total value of the trade. The latter determines
whether unwinding the position would generate a net inflow (i.e. a capital gain) or an
outflow (i.e. a capital loss).
5.8.3.1 The Value of the Reverse Repo Given our assumptions, computing the
value of the reverse repo position at every quarter t is relatively straightforward: Because
we assume that PiVe Capital and the repo dealer simply roll over the $100 million loan
194
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Figure 5.7
The Swap Spread and LIBOR-Repo Spread
2
Swap Spread
LIBOR − Repo Spread
1.5
59 bps
Spread (%)
1
0.5
August
2007
0
June 30, 2006
−0.5
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
Source: Federal Reserve and Bloomberg.
every quarter, the “value” of the repo position on rollover quarter t after the payment of
interest and coupon (if any) is
Value reverse repo at t = $100mm − N × Pn ote (t, T )
where Pn ote (t, T ) is the value of the T-note at time t. This is given by the quoted value
plus accrued interest. Between rollover quarters, a little adjustment has to be made to the
value of the $100 million loan, as by closing the position PiVe Capital would receive any
accrued interest on the repo position, which depends on the repo rate determined on the
last rollover date.
The dotted line in Panel A of Figure 5.8 shows the value of the reverse repo. After June,
2006 the T-note increased in value quite substantially, pushing the value of the reverse repo
down into negative territory. As mentioned, under these circumstances the repo dealer
would likely request PiVe Capital to post additional collateral. The dotted line can then be
interpreted simply as the additional collateral that PiVe Capital has to post over time.
5.8.3.2 The Value of the Fixed-for-Floating Swap The value of the swap at
initiation is zero, as the swap rate c = 5.69% is the rate that makes the swap value zero.
However, as time passes and interest rates fluctuate, the value of the swap changes as well,
as discussed in Section 5.4.1. In particular, from Equation 5.39 we have that the value of
the swap at any time t is given by
Value swap at t = PF R (t, T ) − Pc (t, T )
where PF R (t, T ) is the value of a quarterly floating rate bond at time t with maturity date
T , and quarterly coupons equal to the 3-month LIBOR. Pc (t, T ) is the value of a fixed
CASE STUDY: PIVE CAPITAL SWAP SPREAD TRADES
195
rate coupon bond with maturity T and semi-annual coupons equal to the swap rate, namely
c = 5.69%.
In Section 2.5 in Chapter 2 we discuss how to price floating rate bonds. In particular,
recall that at reset dates PF R (t, T ) = $100, while between reset dates we must also consider
the present value of the next coupon, now fixed at the LIBOR determined on the past reset
date. The fixed coupon bond can be computed using the usual formula
n
Pc (t, T ) =
t
100 × 5.69%
×
Z(t, Ti ) + 100 × Z(t, Tn t )
2
i= 1
where nt is the number of payments remaining at time t, Ti are the payment dates, and
Z(t, Ti ) is the discount curve. The last issue is what discount curve Z(t, Ti ) should we
use? Because we are pricing a swap, using the swap curve appears the most reasonable
methodology. To do so, every t we can recompute the discount curve Z(t, Ti ) from the
quoted swap rates. In particular, we use the bootstrap procedure in Equations 5.44 and
5.45. More specifically, we use data on LIBOR rates for maturities up to 6 months, and
then, given Z(t, t + 0.5), we can apply Equation 5.45 to obtain the curve from quoted
swap rates. Figure 5.9 plots the term structure of interest rates from June 30, 2006 to June
30, 2008. As can be seen, the LIBOR curve was essentially flat on June 30, 2006, but it
dropped dramatically in August 2007 and especially in January 2008.
The solid line in Panel A of Figure 5.8 plots the value of the swap over time. Because
PiVe Capital is long the coupon bond implicit in the swap, the value of the swap increases
over time, as the (average) yield curve decreases. If at any point PiVe Capital decides to
close the swap position, the solid line represents the inflow of money that it would receive
at that time.
5.8.3.3 Aggregate Value As mentioned in the previous section, in December 2007,
PiVe Capital would suffer the first big cash flow decrease. This cash flow is still positive,
as the semi-annual coupon spread remains higher than the quarterly LIBOR-repo spread,
but it is much smaller than in the past. Note that the cash flow realized in December 2007
depends on quantities determined in September 2007. Thus, PiVe Capital could forecast
this shortfall three months in advance. If it wanted to get out of the position, it would have
to close the swap and close the repo transaction. Panel B of Figure 5.8 shows the sum of the
swap value and the reverse repo value, which corresponds to the total amount Pive Capital
would receive or pay if it wanted to close the position. In September 2007, this amount is
close to zero. If PiVe Capital were to close the position then, the trade would still yield a
total positive cumulative cash flow of about $311,000 (see the solid line in Figure 5.6).
Unfortunately, the principals of PiVe Capital decided that the credit crisis was going to
be short lived, and the aggressive response of the Federal Reserve would push the LIBORrepo spread back in line soon enough to keep the positive cash flows going forward. And
if they needed to, they could always unwind the position later.
As we know from the previous section, the LIBOR-repo spread failed to converge back to
reasonable numbers, and PiVe Capital started paying large amount of money every quarter.
It is now March 2008, and the principals decide that it is better to unwind the swap spread
trade positions. The cumulative cash flows still show a positive amount, so not all is lost.
5.8.3.4 The Surprise As PiVe Capital started unwinding the position, the principals
found the good news that the value of the swap was now $10.21 million. Indeed, the LIBOR
196
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
yield curve now ranges between 2.7% at 1 month, and 3.3% at 5 years, which pushes up the
value of the fixed part of the swap to $110.21 million. Because the floating part is always
equal to $100 million at reset dates, by closing the position PiVe Capital would obtain an
inflow of $10.21 million.
However, as PiVe Capital closes the swap, it has also to close the reverse repo. The
bad news is that now the T-note that PiVe is shorting is trading at $110.23. Given the
accrued interest of $1.28, the total losses from the short position amount to $100 million
−N × $111.51 = −$11.38 million.
In sum, to close the position on March, 2008 PiVe Capital receives $10.21 million from
the swap, but it has to pay $11.38 million on the reverse repo, for a total loss of $1.16
million. This number far outweights the cumulative cash flows so far received from PiVe
Capital.
Panel B of Figure 5.8 shows the total value of the trade over time, and indeed, starting
at November 2007, the value has been consistently negative. Why is the value of the swap
spread trade negative? To understand what happened, we have to look again at Figure
5.7. As can be seen from the solid line, the swap spread increased substantially around
November 2007. This implies that the yield of Treasuries decreased compared to the swap
rates. This change is not surprising, as during crisis periods investors purchase safe U.S.
Treasury securities, pushing their yields down. The relative decrease of the yield increased
the value of the Treasury note by more than the increase in the fixed rate bond implicit
in a swap. Because in this swap spread trade PiVe Capital was long the bond implicit in
the swap and short the Treasury note, an increase in the swap spread implies a decrease
in the value of the position, that is, a capital loss. Indeed, note that initially, from June
2006 to June 2007, the swap spread actually decreased over time. Accordingly, Panel B in
Figure 5.8 shows that the value of the swap spread position increased in the first year of
the trade. Had PiVe Capital unwound the position around December 2006, it would have
made $170,974 total from the cash flow side, and about $756,341.33 as a capital gain from
closing the positions, for a total of $927,315.76 within the six months from inception of
the trade. Unfortunately, PiVe Capital didn’t close the position, and suffered a capital loss
instead.
5.8.4 Conclusion
This case study illustrates the risks embedded in fixed income, relative value trades. There
are two types of risk in such trades: The first source of risk is the variation in the cost-ofcarry, which is the cash-flow per period that a trade is expected to generate. In the case of
a swap spread trade, it is the difference between the swap spread and the LIBOR-repo rate.
Typically this difference is positive, implying that setting up the trade generates a positive
cash flow. However, floating rates can move, and thus we have to take into account the fact
that the cash flow can turn negative at times, as it did in 2007 - 2008. The second source
of risk is given by the potential capital losses due to the variation in the underlying spread.
Indeed, in a relative value trade speculators bet on the convergence of the underlying spread
to an average level. In the case study, the swap spread was high in 2006, although not very
high, and PiVe Capital bet that the spread would narrow. If the swap spread narrowed, then
the hedge fund would make substantial capital gain profits. However, even for those trades
in which a hedge fund can reasonably expect that sooner or later the spread will narrow, the
hedge fund manager must keep in mind that it is possible that a spread may widen further
CASE STUDY: PIVE CAPITAL SWAP SPREAD TRADES
197
Figure 5.8 The Swap Spread Trade Value
Panel A. Value of Swap and Reverse Repo
Value (Millions USD)
15
10
Swap
Reverse Repo
5
0
−5
−10
−15
06/30/06 09/30/06 12/31/06 03/31/07 06/30/07 09/30/07 12/31/07 03/31/08 06/30/08
Panel B. Total Value of Trade
Value (Millions USD)
1
0.5
0
−0.5
−1
−1.5
−2
06/30/06 09/30/06 12/31/06 03/31/07 06/30/07 09/30/07 12/31/07 03/31/08 06/30/08
198
INTEREST RATE DERIVATIVES: FORWARDS AND SWAPS
Figure 5.9
The LIBOR Curve: June 2006 – June 2008
6
5.5
5
Yield (%)
4.5
4
3.5
3
2.5
2
5
4
06/30/2006
3
12/31/2006
2
Time to Maturity
06/30/2007
1
12/31/2007
0
06/30/2008
before it narrows. If the spread widens, then the hedge fund would sustain potentially
large capital losses. It is then key for the hedge fund’s survival that it has sufficient capital
in cash or a large borrowing capacity that makes it able to withstand the capital loss.
During periods of crisis, all risk-based spreads – the spreads between a risky securities
and Treasury securities – tend to widen, as investors dump risky securities and purchase
safe U.S. Treasuries. This flight-to-quality indeed took place during the credit crisis of
2007 - 2008, as spreads across a wide spectrum of abitrage strategies widened substantially.
CHAPTER 6
INTEREST RATE DERIVATIVES: FUTURES
AND OPTIONS
6.1
INTEREST RATE FUTURES
Futures contracts are quite similar to forward contracts (see Chapter 5), as they also are
contractual agreements between two counterparties to deliver a certain security (or cash) at
a predetermined time in the future and for a predetermined price, called the futures price.
However, futures contracts differ from forward contracts in a few key respects:
1. The futures contract is traded on a regulated exchange, such as the Chicago Board of
Trade (CBOT) or the Chicago Mercantile Exchange (CME). The exchange defines
the characteristics of the contract and it acts as a counterparty to investors who want
to go long or short the contract. In addition, it also guarantees that payments will be
honored at maturity, through the exchange clearinghouse.
2. The security underlying the futures contract is standardized, in the sense that the
futures contract clearly specifies the type of security that is eligible for delivery, as
well as the time and the method of delivery of the security. That is, there is no room
for customized requests from clients.
3. The profits and losses are marked-to-market daily, meaning that they accrue over
time to short and long traders with daily frequency.
199
200
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
Table 6.1 Some Futures Contracts
Futures Contract
Exchange
30-year U.S. Treasury bond futures
2-, 5-, and 10-year U.S. Treasury note futures
5-, 10-, and 30-year interest rate swap futures
30 day Federal Funds
Eurodollar
LIBOR
2-, 5-, 10-year interest rate swap futures
13-week T-bill
Euribor
Eurobund
CBOT
CBOT
CBOT
CBOT
CME
CME
CME
CME
Euronext
Eurex
Interest rate futures are available on numerous exchanges and on numerous instruments.
Table 6.1 provides a partial list of interest rate futures, and the exchanges on which they
trade.
6.1.1 Standardization
The 10-year, U.S. Treasury note futures, whose contract characteristics are contained in
Table 6.2, provide a clear example of standardization. This futures contract trades on the
Chicago Board of Trade, and it is among the most traded and liquid futures contract in the
world. The standardization is clearly shown in the items “deliverable grade” and “contract
months.” The deliverable grade item specifies in detail the type of Treasury note that is
eligible for delivery. To enhance liquidity and trading in the futures contract, the exchange
specifies a large range of possible Treasury notes that are eligible for delivery, specifically,
all those Treasury notes that have at least 6 1/2 years to maturity, but no more than 10.
The short side of the futures contract, i.e. the counterparty who commits to delivering
the security, can then choose any of these securities to deliver to the long side, i.e. the
counterparty who commits to purchasing the security. To make these securities comparable
with one another, the futures price at maturity will be multiplied by a conversion factor that
(almost) eliminates the differences in bond prices due to differences in coupon rates.1
Similarly, the contract months item defines the possible maturities of the futures contract.
By establishing only four expiration months, and even a specific period in the month,
the exchange (CBOT) can enhance market liquidity. Indeed, if many more maturities
were available, the number of contracts traded per maturity would be smaller, and market
liquidity would suffer. As explained below, however, substantially decreasing the number
of available maturities has an important drawback as it increases the maturity mismatch
between the available contract maturities and the hedging needs of end users of the futures
contracts. If the maturity mismatch becomes too large, then corporations may opt out
1 If the futures contract was defined on a specific Treasury note, such as the one that is closest to 8 years to maturity
at delivery time, then some investors might “squeeze” the short side by buying a large amount of available 8 years
notes. Because the short side must deliver at maturity, it will be willing to pay a large premium to just get hold
of the note to deliver. This possibility would break the futures market itself, as expecting this behavior, investors
would not enter into the short side of a futures contract to start with.
INTEREST RATE FUTURES
201
Table 6.2 10-Year U.S. Treasury Note Futures (Chicago Board of Trade)
Contract Size
One U.S. Treasury bond having a face value at maturity of $100,000 or multiple thereof.
Deliverable Grades
U.S. Treasury notes maturing at least 6 1/2 years, but not more than 10 years, from the first day of
the delivery month. The invoice price equals the futures settlement price times a conversion factor
plus accrued interest. The conversion factor is the price of the delivered note ($1 par value) to
yield 6 percent.
Tick Size
Minimum price fluctuations shall be in multiples of one-half of one thirty-second (1/32) point per 100
points ($15.625 rounded up to the nearest cent per contract).
Price Quote
Points ($1,000) and one half of 1/32 of a point; i.e., 84-16 equals 84 16/32, 84-165 equals 84 16.5/32.
Contract Months
Mar, Jun, Sep, Dec
Last Trading Day
Seventh business day preceding the last business day of the delivery month. Trading in expiring
contracts closes at noon, Chicago time, on the last trading day.
Last Delivery Day
Last business day of the delivery month.
Delivery Method
Federal Reserve book-entry wire-transfer system.
Trading Hours
Open Auction: 7:20 am - 2:00 pm, Central Time, Monday - Friday
Electronic: 6:00 pm - 4:00 pm, Central Time, Sunday - Friday
Ticker Symbols
Open Auction: TY
Electronic: ZN
Daily Price Limit
None
Margin Information
Initial Margin: $1,890 (per contract)
Maintenance Margin: $ 1,400 (per contract)
Source: CBOT Web site, http://www.cbot.com/cbot/pub/cont detail/1,3206,1520+14433,00.html,
accessed on June 11, 2008.
from the futures market in favor of less liquid, but highly customized over-the-counter
forward contracts. The exchange must then weigh the need of corporations to have many
maturities per year available for hedging purposes against the decrease in liquidity if too
many maturities were in fact available.
The 10-year U.S. Treasury note futures and the 30 year U.S. Treasury bond futures have
numerous peculiarities which we will study in more detail in Chapter 11.
202
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
6.1.2 Margins and Mark-to-Market
One essential feature of futures contracts is marking-to-market, that is, the profits and
losses from the futures contract trading activity accrue to traders with daily frequency. To
understand the logic, it is useful to think about forward contracts, described in Chapter 5.
Consider a forward contract to buy at time T a given coupon bond with maturity T ∗ > T
(see Fact 5.9 in Chapter 5). As we know, the value of entering into a forward contract is zero
at initiation t = 0. Now, let one day pass by. As interest rates move, so does the forward
price, and from Equation 5.36 the value of the forward contract at t = dt = 1/252 = 1
day, is
#
$
V f w d (dt) = Z(dt, T ) × Pcf w d (dt, T, T ∗ ) − Pcf w d (0, T, T ∗ )
(6.1)
Because it did not cost anything to enter into the forward contract, the amount V f w d (dt)
is the daily profit or loss: For instance, if the forward price increases, then the forward
contract gains in value, and vice versa. In a typical forward contract, there is no exchange
of money between the two counterparties as V f w d (t) changes from one day to the next.
However, in principle, the two counterparties can decide to mark-to-market the forward
contract, meaning that every day the party with negative V f w d (dt) pays this amount to
the other one, thereby resetting the value of the forward contract to zero. The profits and
losses from the forward position accrue over time to the counterparties. This practice
is very popular across dealers and investment banks as it limits the credit risk exposure
from derivative transactions. Indeed, from the discussion following Equation 5.36 daily
mark-to-market is equivalent to the strategy calling for the daily closing of the old forward
contract [with forward price Pcf w d (t − dt, T, T ∗ )] and opening a new one [with forward
price Pcf w d (t, T, T ∗ )]. Thus, the profit / loss at any time t is given by
#
$
Profit / Loss at t = V f w d (t) = Z(t, T )× Pcf w d (t, T, T ∗ ) − Pcf w d (t − dt, T, T ∗ ) (6.2)
We now see that receiving the sequence of payments V f w d (t) at daily frequencies, that
is, times t = dt, 2 × dt, 3 × dt, .... T , is equivalent to the original payoff from the forward
contract, namely, Equation 5.29. The important point to notice is that the sequence of cash
flows V f w d (t) occur at time t, while the final payoff is at maturity T . Thus, we have to
take the future value of each of these cash flows from t to T . To do so we must multiply
each of the cash flows at time t in Equation 6.2 by 1/Z(t, T ). We then obtain
Total profits / losses at T
=
V f w d (dt) V f w d (2 × dt)
+
+ ... + V f w d (T ) (6.3)
Z(dt, T )
Z(2 × dt, T )
From Equation 6.2, we see that for every t,
$
V f w d (t) # f w d
= Pc (t, T, T ∗ ) − Pcf w d (t − dt, T, T ∗ ) ,
Z(t, T )
which implies
Total profits / losses at T
=
$
# f wd
Pc (dt, T, T ∗ ) − Pcf w d (0, T, T ∗ )
$
#
+ Pcf w d (2 × dt, T, T ∗ ) − Pcf w d (dt, T, T ∗ )
INTEREST RATE FUTURES
203
..
.
$
#
+ Pcf w d (T, T, T ∗ ) − Pcf w d (T − dt, T, T ∗ )
$
#
= Pc (T, T ∗ ) − Pcf w d (0, T, T ∗ )
where the last equality stems from eliminating all of the common terms [e.g., “Pcf w d (dt, T, T ∗ )”
in the first row cancels with “−Pcf w d (dt, T, T ∗ )” in the second row, and so forth], and the
fact that at maturity, the forward price equals the price of the bond Pcf w d (T, T, T ∗ ) =
Pc (T, T ∗ ). We have established the following:
Fact 6.1 In a forward contract, marking-to-market does not change the final payoff to
either counterparty.
Futures markets work exactly in this fashion: Every day the futures price, P f u t (t, T ),
moves and thus profits and losses are credited or debited to the traders’ accounts with the
exchange. In particular, if one trader is long k futures contracts at t, the profit or loss at the
end of trading day t is
$
#
(6.4)
P&L from futures at t = k × Contract size × P f u t (t, T ) − P f u t (t − dt, T )
To enter into a futures position, a trader must post an initial amount of money in a specific
account with the exchange, called the initial margin. For instance, Table 6.2 shows that
the initial margin for the 10-year, U.S. Treasury note futures is $1,890 per contract. As the
futures price moves, the margin account is debited or credited, depending on the movement.
If the total amount in the account declines below the maintenance margin, the exchange
issues a margin call and the trader must replenish the trading account back to the initial
margin. If the trader fails to do so, the futures contract is closed.
Marking-to-market and the requirement for traders to keep a minimum amount of money
as a collateral to the trade limits the credit risk that the exchange takes from each trader.
6.1.3
The Convergence Property of Futures Prices
Interest rate futures prices are harder to analyze than forward prices, so we will discuss
them in more detail after we introduce some modeling devices in Chapter 11. However, one
property that futures prices have in common with forward prices is the fact that they must
converge to the value of the underlying security (or cash amount, if cash settled) by maturity
date. That is, futures prices P f u t (t, T ) on a security with value P (t) at time t have the
property that at maturity P f u t (T, T ) = P (T ). This fact makes it almost equivalent to using
forward contracts or futures contracts for some hedging purposes, at least in principle. The
next example illustrates the similarities, and the differences, between futures and forward
contracts for hedging purposes.
EXAMPLE 6.1
Consider again Example 5.3 in Chapter 5. On March 1, 2001 (today = time 0), a firm
is worried that the interest rate may decline in the next six months, when it will receive
the payment from the sale of a piece of equipment. In particular, the firm will receive
on September 4, 2001 a $100 million receivable that needs to be invested for another
six months. Instead of entering into a forward rate agreement (FRA) as in Example
204
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
5.3, the firm may decide to use futures contracts to hedge against a decline in interest
rates. For instance, the firm may go long k Eurodollar futures contracts. Table 6.3
contains the details of the Eurodollar futures contract. In particular, although the
variable underlying the futures is a rate (the 3-month LIBOR), the quote is in prices
given by P F u t (t, T ) = 100 − f4F u t (t, T ), where f4F u t (t, T ) is the 3-month futures
LIBOR rate at T , expressed as a percentage. For instance, the quote P F u t (t, T ) =
95.365 implies a rate f4F u t (t, T ) = (100 − 95.365)/100 = 4.635%. In particular,
at maturity, the futures price will be equal to P F u t (T, T ) = 100 − r4L I B O R (T ).
This quoting convention implies that a long position in the Eurodollar futures hedges
against a decline in interest rates. Given the contract size of $1 million, the daily
profits and losses from the futures contract are then given by
F ut
P
(t + dt, T ) − P F u t (t, T )
(6.5)
Daily P&L = $1, 000, 000 × 0.25 ×
100
F ut
= $1, 000, 000 × 0.25 × f4 (t, T ) − f4F u t (t + dt, T )
(6.6)
where 0.25 reflects the fact that the Eurodollar futures is a 90-day interest rate, and
dt = 1/252 =1 day.
On March 1, 2001, the September, 2001 futures contract was quoted at f4F u t (0, T1 ) =
4.635%, where T1 = September 17, 2001 is the futures maturity. To hedge the $100
million position, the firm must go long 100 Eurodollar futures contracts. The futures
rate on September 4, 2001 (= t) was f4F u t (t, T1 ) = 3.520%.2 The total P/L from the
futures contract between March 1 and September 4, 2001, is then:3
Total
profit/losses
from futures
=
100 × $1, 000, 000 × 0.25 × 4.635% − f4F u t (dt, T1 )
+100 × $1, 000, 000 × 0.25 × f4F u t (dt, T1 ) − f4F u t (2dt, T1 )
..
..
.
.
+100 × $1, 000, 000 × 0.25 × f4F u t (t − dt, T1 ) − 3.520%
=
100 × $1, 000, 000 × 0.25 × (4.635% − 3.520%) = $278, 750
Table 6.4 also reports the daily P/L for a selected period of time.
On September 4, 2001 (first business day of the month) the firm receives the $100
million receivable, which it can now invest at the current interest rate for the next
six months. Because the firm is hedging with a LIBOR-based contract, let’s assume
it can invest for six months using a 6-month LIBOR. The six month LIBOR on
September 4, 2001 is r2 (t, t + 0.5) = 3.55%. Thus, the firm’s total payoff at time
T2 = March 1, 2002 is
3.55%
Total amount at T2 = ($100 mil + $278, 750) × 1 +
= $102.059 million
2
that the firm will unravel the position at t = September 4, 2001, which is a little earlier than the maturity
of the futures contract T 1 = September 17, 2001.
3 We are ignoring for now the impact of the timing of cash flows on the final payoff. See discussion in Section
6.1.4 below.
2 Note
INTEREST RATE FUTURES
Table 6.3
205
CME Eurodollar Futures
Underlying Instrument
3-month LIBOR: London Interbank Offered Rate on 3-month U.S. dollar deposits.
Contract Size
$1,000,000
Tick Size
Trading can occur in .0025 increments ($6.25/contract) in the expiring front-month contract;
in .005 increments ($12.50/contract) in the four serial and all forty quarterly expirations.
Price Quote
P F u t (t) = 100 − f4F u t (t), where f4F u t is the 3-month “expected” LIBOR rate.
Contract Months
Mar, Jun, Sep, Dec, extending out 10 years (total of 40 contracts) plus the four nearest serial
expirations (months that are not in the March quarterly cycle).
Last Trading Day
Seventh business day preceding the last business day of the delivery month. Trading in expiring
contracts closes at noon, Chicago time, on the last trading day.
Final Settlement
Cash settlement to the British Bankers Association survey of 3- month LIBOR.
Trading Hours
Open Outcry: 7:20 a.m. û 2:00 p.m.
CME Globex Electronic Markets: 5:00 p.m. û 4:00 p.m. the following day;
on Sunday, trading begins at 5:00 p.m.
Source: CME Interest Rate Product Guide and Calendar, 2007.
This number is not too dissimilar from the one obtained using the forward contract,
or the forward rate agreement, namely, $102.105 million.
Figure 6.1 plots the September, 2001 Eurodollar futures and the 3-month LIBOR.
The convergence property is apparent.
6.1.4
Futures versus Forwards
In this subsection we discuss the relation between futures and forwards, as well as the pros
and cons in using them for hedging purposes.
First of all, what is the relation between futures prices and forward prices? They are
very closely related. In particular, consider a forward contract to receive at time T1 a zero
coupon bond Pz (T1 , T2 ) for the delivery price (or forward price) Pzf w d (0, T1 , T2 ). The
payoff at maturity T1 of this security is
Forward contract payoff at T1
= Value of zero coupon bond − Delivery price
= Pz (T1 , T2 ) − PzF w d (0, T1 , T2 )
Consider now a futures contract that is otherwise analog to the forward contract above.
Because of mark-to-market, every trading day t the daily P/L from a short position in
206
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
Table 6.4
Profits and Losses from Futures Position
Date
Quote
Rate
1-Mar-01
2-Mar-01
5-Mar-01
6-Mar-01
7-Mar-01
8-Mar-01
9-Mar-01
12-Mar-01
13-Mar-01
14-Mar-01
15-Mar-01
16-Mar-01
19-Mar-01
20-Mar-01
21-Mar-01
22-Mar-01
23-Mar-01
26-Mar-01
27-Mar-01
28-Mar-01
29-Mar-01
30-Mar-01
..
.
1-Aug-01
2-Aug-01
3-Aug-01
6-Aug-01
7-Aug-01
8-Aug-01
9-Aug-01
10-Aug-01
13-Aug-01
14-Aug-01
15-Aug-01
16-Aug-01
17-Aug-01
20-Aug-01
21-Aug-01
22-Aug-01
23-Aug-01
24-Aug-01
27-Aug-01
28-Aug-01
29-Aug-01
30-Aug-01
31-Aug-01
4-Sep-01
95.3650
95.3200
95.3200
95.3500
95.3900
95.4050
95.3400
95.3650
95.3650
95.5250
95.6250
95.6000
95.5550
95.6350
95.7000
95.7650
95.6700
95.6700
95.5200
95.5700
95.6200
95.6900
..
.
96.4650
96.4300
96.4300
96.4300
96.4300
96.4850
96.5050
96.5350
96.5475
96.5325
96.5000
96.5375
96.5650
96.5450
96.5650
96.5475
96.5475
96.5250
96.5200
96.5525
96.5575
96.5900
96.5700
96.4800
4.635%
4.680%
4.680%
4.650%
4.610%
4.595%
4.660%
4.635%
4.635%
4.475%
4.375%
4.400%
4.445%
4.365%
4.300%
4.235%
4.330%
4.330%
4.480%
4.430%
4.380%
4.310%
..
.
3.535%
3.570%
3.570%
3.570%
3.570%
3.515%
3.495%
3.465%
3.453%
3.468%
3.500%
3.463%
3.435%
3.455%
3.435%
3.453%
3.453%
3.475%
3.480%
3.448%
3.443%
3.410%
3.430%
3.520%
Data Source: Bloomberg.
Daily P/L
Cumulative P/L
-112.50
0.00
75.00
100.00
37.50
-162.50
62.50
0.00
400.00
250.00
-62.50
-112.50
200.00
162.50
162.50
-237.50
0.00
-375.00
125.00
125.00
175.00
..
.
0.00
-87.50
0.00
0.00
0.00
137.50
50.00
75.00
31.25
-37.50
-81.25
93.75
68.75
-50.00
50.00
-43.75
0.00
-56.25
-12.50
81.25
12.50
81.25
-50.00
-225.00
-112.50
-112.50
-37.50
62.50
100.00
-62.50
0.00
0.00
400.00
650.00
587.50
475.00
675.00
837.50
1,000.00
762.50
762.50
387.50
512.50
637.50
812.50
..
.
2,750.00
2,662.50
2,662.50
2,662.50
2,662.50
2,800.00
2,850.00
2,925.00
2,956.25
2,918.75
2,837.50
2,931.25
3,000.00
2,950.00
3,000.00
2,956.25
2,956.25
2,900.00
2,887.50
2,968.75
2,981.25
3,062.50
3,012.50
2,787.50
INTEREST RATE FUTURES
Figure 6.1
207
Eurodollar Futures and the three month LIBOR
5.5
Eurodollar Futures
3−month LIBOR
5
Rate (%)
4.5
4
3.5
3
2001/04/01
2001/07/01
2001/09/17
Source: Bloomberg.
futures is given by (see Equation 6.4)
#
$
Daily P&L from short futures = (P/L)t = Pzf u t (t, T1 , T2 ) − Pzf u t (t − dt, T1 , T2 )
(6.7)
Ignoring for now the time value of money, the total payoff at time T1 of a short position is
given by the sum of the daily profits and losses:
Total P/L of short futures at T1
=
(P/L) + (P/L)2dt + .... + (P/L)T
(6.8)
# f u t dt
$
f ut
= Pz (0, T1 , T2 ) − Pz (dt, T1 , T2 )
$
#
+ Pzf u t (dt, T1 , T2 ) − Pzf u t (2 × dt, T1 , T2 )
..
.
$
#
+ Pzf u t (T1 − dt × dt, T1 , T ) − Pzf u t (T1 , T1 , T2 )
#
$
= Pzf u t (0, T1 , T2 ) − Pz (T1 , T2 )
It follows that if we enter into a forward contract at time 0 and hedge it with a futures
position, we obtain that the total payoff is given by the difference between futures and
forward prices:
Long forward + Short futures payoff
=
$
#
Pz (T1 , T2 ) − PzF w d (0, T1 , T2 )
$
#
+ Pzf u t (0, T1 , T2 ) − Pz (T1 , T2 )
= Pzf u t (0, T1 , T2 ) − PzF w d (0, T1 , T2 )
208
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
Given that both the futures price and the forward price are known at time 0 and the fact
that it costs nothing to enter into either a forward or a futures, we must have the no arbitrage
condition
No arbitrage =⇒ Pzf u t (0, T1 , T2 ) − PzF w d (0, T1 , T2 ) = 0
We therefore obtain the following:
Fact 6.2 Under the following two assumptions:
1. we ignore the difference in timing of cash flows between a forward or a futures
contract; and
2. the futures and forward contract payoff occur on the same date (T1 );
then the futures price must equal the forward price:
Pzf u t (0, T1 , T2 ) = PzF w d (0, T1 , T2 )
(6.9)
Equation 6.9 is obtained under assumptions 1 and 2 in Fact 6.2. Both assumptions are
violated in practice: First, futures do in fact generate cash flows over time, a fact that makes
Equation 6.8 incorrect. Instead, the correct expression is one that includes the future value
of each daily P/L. Second, there is often a mismatch between forward and futures payoff
timing: For instance, while a plain vanilla Forward Rate Agreement on the three month
LIBOR would pay the amount N × Δ × [f4 (0, T1 , T2 ) − r4 (T1 , T2 )] at T2 = T1 + Δ, the
Eurodollar futures contract pays the same payoff at T1 .
These differences slightly change the relation between futures prices and forward prices.
However, the difference between forward prices and futures prices is small, and thus
Equation 6.9 is a good approximation, especially if the maturity T is not large and the
volatility of interest rates is also small. We explore the exact relation between futures and
forwards in Chapter 21. Section 6.5 below provides additional insights on the relation
between futures and forward prices.
6.1.5 Hedging with Futures or Forwards?
Example 6.1 highlights two possible shortcomings of futures versus forward:
1. Basis Risk. The available maturity of the bond, or the particular instrument may
not be the exact instrument to hedge all of the risk. Using a forward rate agreement,
a firm could perfectly hedge the risk. Using futures, the firm would retain some
residual risk, as the available instruments (the Eurodollar futures, based on the 3month LIBOR) is not perfectly correlated with the interest rate to hedge (which is a
6-month rate). Moreover, there is a (mild) maturity mismatch between the needs of
the firm and the available maturities, as the Eurodollar futures expire in the middle
of September, while the receivable arrives at the beginning of September.
2. Tailing of the Hedge. The cash flows arising from the futures position accrue over
time, which implies the need of the firm to take into account the time value of money
between the time at which the cash flow is realized and the maturity of the hedge
OPTIONS
209
position (maturity T in the example). This will call for a reduction in the position in
futures, compared to the description in the example. We follow up on this notion in
Exercise 9.
On the other hand, futures also have numerous advantages compared to forwards, which
may more than compensate for the two problems above.
1. Liquidity. Because of their standardization, futures are more liquid than forward
contracts, meaning that it is easy to get in and out of the position.4 For the highly
traded futures contracts, such as the 10-year U.S. Treasury note futures or the Eurodollar futures, bid/ask spreads are relatively low and going in and out of positions
is relatively inexpensive. This is not true for forward contracts, because as they are
traded only over-the-counter, closing a position may be expensive.
2. Credit Risk. The existence of a clearinghouse guarantees performance on futures
contracts, while the same may not be true for forward contracts. The clearing house
hedges itself through the mark-to-market provision: As soon as one trader’s account
falls below the margin requirements, a margin call is issued and, if the account is not
replenished, the position is closed. This mechanism guarantees to some extent that
no large credit exposure is mounted for any single trader.
6.2
OPTIONS
All of the derivative contracts discussed so far, namely forward rate agreements, forward
contracts, swaps and futures, share two common features:
1. It costs nothing to enter in such derivative contracts;
2. Either counterparty may be called to make a payment at maturity.
For instance, a forward rate agreement costs nothing at inception, but the payoff at maturity
T2 of the FRA is
Payoff of FRA at T2 = N × Δ × [rn (T1 , T2 ) − fn (0, T1 , T2 )]
where rn (T1 , T2 ) is the reference floating rate at T1 , fn (0, T1 , T2 ) is the corresponding
forward rate at 0, and Δ = T2 − T1 = 1/n is the compounding interval. The payoff to
either party at maturity can be both positive or negative.
Options are different in both respects. First, there is an exchange of money when the two
counterparties enter into an option contract: One counterparty, the option buyer, pays an
amount of money, the option premium, to the other counterparty, the option seller. Second,
at maturity, the option buyer never makes a positive net payment, while the option seller
may suffer a net outflow under some circumstances that depend on the underlying variable.
The variety of interest rate options is very large, and in the next few subsections we
describe the most popular. First, it may be useful to describe a few generic characteristics
that are common across the various types of options. The next definition introduces some
of the terminology, and the effective option’s payoff.
4 The
level of liquidity really depends on the futures contract, as some are in fact quite illiquid, with large bid/ask
spreads.
210
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
Definition 6.1 A call option defined on the variable F (t), with maturity T and strike
price K, is a contract between two counterparties, the option buyer and the option seller,
according to which:
(A) Any time t prior to maturity, the option buyer has the right to ask the option seller
for the payment of the following effective payoff:
Call option payoff = max(F (t) − K, 0)
(6.10)
(B) The option seller has the obligation to pay this amount to the option buyer at t.
(C) In return for the right to obtain the payment in Equation 6.10, the option buyer pays
an option premium to the option seller at time 0.
The act of requesting the payment in Equation 6.10 is called exercising the option. If
the option can be exercised only at maturity T , then the option is termed European. If the
option can be exercised any time before maturity, it is called American.5
Finally, a put option is the same as above, except that its payoff is
Put Option Payoff = max(K − F (t), 0)
(6.11)
Definition 6.1 is generic, and it describes the effective payoff from an option contract.
This payoff may arise in different ways, depending of the nature of the option.
1. Bond Options. The underlying variable in a bond option is a fixed coupon instrument,
such as a U.S. Treasury note. In this case, the underlying variable in Definition 6.1
is the coupon bond price F (t) = Pc (t, TB ), where TB is the maturity of the coupon
bond, with TB > T . A call option contract then specifies that the option’s buyer has
the right, but not the obligation, to purchase the underlying security any time before
or at maturity T for a price K set at initiation. Because the option buyer will never
exercise the option if Pc (t, TB ) < K – it is better to purchase the coupon bond in
the cash market in this case – effectively the option buyer is entitled to the payoff
described in Equation 6.10. The option seller in turn must effectively make the
payment in Equation 6.10, as he or she must deliver a coupon bond valued Pc (t, TB )
but only receives the strike price K < Pc (t, T ).
The inverse argument holds for put options: In this case, the option buyer has the
right, but not the obligation, to sell the coupon bond at Pc (t, TB ) for a price K.
Because again the option buyer will not exercise the option unless K > Pc (t, TB ),
the option buyer is effectively entitled to the payoff in Equation 6.11. The option
seller has the obligation to purchase the coupon bond at a higher price (K) than the
market price, and therefore suffers a loss.
2. Interest Rate Options. The underlying variable is an interest rate, such as the 13week Treasury discount rate, or the 3-month LIBOR. In this case, F (t) in Definition
6.1 is an interest rate, such as F (t) = r4 (t), and the strike price K is in fact a strike
5 The
terms “American” and “European” have nothing to do with the location where the options are traded.
OPTIONS
211
“rate.” These options are cash settled. Table 6.5 shows the contract details, for
instance, of the IRX option on the 13-week discount rate listed at the CBOE.
A class of popular interest rate options that are traded in the over-the-counter market
is caps and floors. A quarterly cap with maturity T and strike rate K is a security
that pays at times T1 , T2 , ... Tn , with Ti = Ti−1 + 0.25 the sequence of cash flows
Payoff of a cap at Ti = N × 0.25 × max(r4 (Ti−1 ) − K, 0)
where the reference rate r4 (t) is the 3-month LIBOR rate. A floor, instead pays the
amount N × 0.25 × max(K − r4 (Ti−1 ), 0). A cap, therefore, is given by a portfolio
of n interest rate options.
3. Futures Options. The underlying is the futures price of another contract, such as the
10 year, U.S. T-note futures. In this case, the variable in Definition 6.1 is a futures
price F (t) = F f u t (t, TF ), where TF is the maturity of the futures. A call option
contract then specifies that the option’s buyer has the right, but not the obligation, to
enter into a long futures position any time before or at maturity T for a futures price
K set at initiation. Exchanges that list futures contracts typically also list options on
their own futures contracts, as we can see from Table 6.6. Tables 6.7 and 6.8 show the
terms of the contracts for the 10-year, U.S. T-note futures option, and the Eurodollar
futures option, respectively. The description of the futures contracts themselves are
in Tables 6.2 and 6.3.
4. Swaptions.
The underlying is a swap. In this case, the underlying variable in
Definition 6.1 is the current swap rate F (t) = c(t, T ), where T is the maturity of the
swap, and the strike price K is a strike “swap rate.” A call option contract, or payer
swaption, then specifies that the option buyer has the right, but not the obligation,
to enter into a swap and pay only the strike rate K, instead of the market rate, for
the life of the swap. Clearly, an option buyer would exercise the option only if the
current swap rate is higher than the strike price, c(t, T ) > K, otherwise it is better
to let the option expire and enter into a swap as a fixed payer at the market rate. As
we shall see, the payoff for this option is as in Equation 6.10. A put option contract,
or a receiver swaption, instead specifies that the option buyer has the right to enter
into a swap and receive the strike price K instead of the market rate. Again, in this
case an option buyer would exercise the option only if the strike rate K is above the
market rate, that is, if K > c(t, T ), leading to a payoff of the form in Equation 6.11.
The next subsections provide some examples. Before we turn to these examples, it is
useful to introduce additional terminology.
Definition 6.2 An option is said to be in-the-money (ITM) when it is profitable for the
option’s buyer to exercise it. In particular, a call option is in-the-money if F (t) > K while
a put option is in-the-money if F (t) < K.
An option is said to be out-of-the-money (OTM) when it is not profitable to exercise
it. In particular, a call option is out-of-the-money if F (t) < K while a put option is
out-of-the-money if F (t) > K.
An option is said to be at-the-money (ATM) when F (t) is (approximately) equal to the
strike price K.
212
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
Table 6.5
CBOE 13-week Treasury Bill Option (IRX)
Underlying Variable
10 × the discount rate of the most recently auctioned 13-week U.S. Treasury bill. The new T-bill
is substituted weekly on the trading day following its auction, usually a Monday.
Multiplier
100
Strike Price Intervals
2 1/2 points. A 1-point interval represents 10 basis points.
Premium Quotation
Stated in decimals. One point equals $100. The minimum tick for options
trading below 3.00 is 0.05 ($5.00) and for all other series, 0.10 ($10.00).
Expiration Date
Saturday immediately following the third Friday of the expiration month.
Expiration Months
Three near-term months plus two additional months from the March quarterly
cycle (March, June, September and December).
Exercise Style
European
Settlement of Option Exercise
Annualized discount rate on the most recently issued T-bill on the last trading day as
reported by the Federal Reserve Bank of New York at 2:30 p.m. Central Time.
Exercise will result in delivery of cash on the business day following the expiration date.
The exercise-settlement amount is equal to the difference between the exercise-settlement
value and the exercise price of the option, multiplied by $100.
Last Trading Day
Trading in interest rate options will ordinarily cease on the business day (usually a Friday)
preceding the expiration date.
Trading Hours
7:20 a.m. û 2:00 p.m. (Central Time)
Source: CBOE Web Site.
OPTIONS
213
Table 6.6 Some Exchange Traded Options Contracts
Options Contract
Exchange
Underlying
Exercise
30-year US T-bond
10-year US T-note
5-year US T- note
Eurodollar
LIBOR option
30 day Federal Fund
13-week T-bill
Euroyen TIBOR option
13-week Treasury bill
5-year Treasury note
10-year Treasury note
30-year Treasury note
Euribor
Eurobund
CBOT
CBOT
CBOT
CME
CME
CME
CME
CME
CBOE
CBOE
CBOE
CBOE
Euronext
Eurex
30-year US T-bond futures
10-year US T-note futures
5-year US T-note futures
Eurodollar futures
LIBOR futures
Federal Funds futures
Federal Funds futures
Euroyen futures
13-week T-bill discount rate
Yield on the 5-year T-note
Yield on the 10-year T-note
Yield on the 30-year T-bond
Euribor futures
Eurobund futures
American
American
American
American
American
American
American
American
European
European
European
European
American
American
Figure 6.2 illustrates the payoff diagrams for options on bonds. The payoff for calls and
puts in this case are
max(P (T, TB ) − K, 0)
and
max(K − P (T, TB ), 0),
(6.12)
respectively, where T is the exercise time, K is the strike price, and P (T, TB ) is a reference
bond. Panel A of Figure 6.2 plots the payoff for a call option buyer: If the price of the
bond increases above the strike price (the dotted line), the option buyer gets a positive
payoff, otherwise he or she receives nothing. The diagram also reports the areas commonly
referred to as in-the-money (ITM), out-of-the-money (OTM), and at-the-money (ATM).
The payoff to the call option seller is depicted in Panel B. The payoff diagram is the mirror
image of the call option buyer. Panels C and D show the payoff diagrams for a put option
buyer and a put option seller, respectively.
Because an option buyer pays a premium to the option seller, the payoff is not a profit. To
compute the profit for the option buyer, we must decrease the total payoff by the premium
paid.6 Similarly, to compute the profit to the option seller, we must increase the payoff by
the option premium. Figure 6.3 shows the payoffs (solid line) and the profits (dashed line)
for option buyers and option sellers. Notice that, in particular, an option seller’s profit is at
most the premium obtained for selling the option.
6.2.1
Options as Insurance Contracts
The best way to understand options contracts is to consider them as insurance contracts.
As in any insurance contract, there is a premium to pay up front, and a payment at maturity
that occurs if some adverse event happens. For instance, if you purchase an annual theft
insurance for your car, you pay a premium up front to the insurance company. If your car
6 In
fact, we have to take the future value of the option premium to take into account the time value of money: the
premium is paid at 0 while the payoff at T .
214
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
Table 6.7
10-Year U.S. Treasury Note Option (Chicago Board of Trade)
Contract Size
One CBOT 10-year U.S. Treasury note futures contract (of a specified delivery month) having
a face value at maturity of $100,000 or multiple thereof.
Expiration
Unexercised 10-year Treasury note futures options shall expire at 7:00 p.m. Central Time
on the last day of trading.
Tick Size
1/64 of a point ($15.625/contract) rounded up to the nearest cent/contract.
Contract Months
The first three consecutive contract months (two serial expirations and one quarterly expiration)
plus the next four months in the quarterly cycle (Mar, Jun, Sep, Dec). There will always be seven
months available for trading. Serials will exercise into the first nearby quarterly futures contract.
Quarterlies will exercise into futures contracts of the same delivery period.
Last Trading Day
Options cease trading at the same time as the underlying futures contract on the last Friday
preceding by at least two business days the last business day of the month preceding the option
contract month. Options cease trading at the close of the regular daytime open auction trading
session for the corresponding 10-year Treasury note futures contract.
Last Delivery Day
Last business day of the delivery month.
Trading Hours
Open Auction: 7:20 am - 2:00 pm, Chicago time, Monday - Friday
Electronic: 5:30 pm - 4:00 pm, Chicago time, Sunday - Friday
Ticker Symbols
Open Auction: TC for calls, TP for puts
Electronic: OZN for calls, OZNP for puts
Strike Price Intervals
Strike prices will be listed in integral multiples of one-half of one point ($500 per contract)
to bracket the settlement price of the underlying 10-year U.S. Treasury note futures contract.
Strike prices will included the at-the-money strike price plus the next 50 consecutive higher
and the next 50 consecutive lower strike prices.
Exercise
The buyer of a futures option may exercise the option on any business day prior to expiration
by giving notice to the Board of Trade clearing service provider by 6:00 pm, Central Time.
Options that expire in-the-money are automatically exercised into a position, unless specific
instructions are given to the Board of Trade clearing service provider.
Source: CBOT web site, http://www.cbot.com/cbot/pub/cont detail/0,3206,1520+14438,00.html,
accessed on August 27, 2008.
OPTIONS
215
Figure 6.2 Interest Rate Options Payoff Profiles
B. Call Option Seller
2
2
1
1
Payoff
Payoff
A. Call Option Buyer
0
−1
−1
OTM
−2
95
0
ITM
−2
ATM
96
97
98
Bond Price
99
100
95
96
2
2
1
1
0
−1
100
99
100
0
−1
ITM
−2
95
99
D. Put Option Seller
Payoff
Payoff
C. Put Option Buyer
97
98
Bond Price
OTM
−2
ATM
96
97
98
Bond Price
99
100
95
96
97
98
Bond Price
216
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
Figure 6.3 Option Payoff versus Option Profit
B. Call Option Seller
2
2
1
1
Payoff / Profit
Payoff / Profit
A. Call Option Buyer
0
−1
−2
95
0
−1
−2
96
97
98
Bond Price
99
100
95
96
97
98
Bond Price
99
100
99
100
payoff
2
2
1
1
0
−1
−2
95
D. Put Option Seller
profit
Payoff / Profit
Payoff / Profit
C. Put Option Buyer
0
−1
−2
96
97
98
Bond Price
99
100
95
96
97
98
Bond Price
OPTIONS
Table 6.8
217
CME Eurodollar (Quarterly) Options
Underlying Instrument
The CME Eurodollar futures contract expiring in the same month, and on the same day,
of the option.
Contract Months
The first eight months in the March quarterly cycle (Mar, Jun, Sep, Dec).
Exercise (Strike) Prices
25-basis-point increments, e.g., 95.25, 95.50, 95.75, etc. The two nearest
contract months are eligible for 12.5 basis point increments.
Expiration/Settlement
Cash settle at the same time and on the same day as the underlying futures contract.
Exercise
The option is American style, and it can be exercise any day before maturity up to 7:00 pm
(Chicago time). All in-the-money options are automatically exercised at maturity.
Minimum Price Fluctuation (Tick)
.0025 = $6.25/contract for the nearest expiring option (if underlying futures eligible);
Trading Hours
Open Outcry: 7:20 a.m. û 2:00 p.m.
CME Globex Electronic Markets: 5:00 p.m. û 4:00 p.m. the following day;
on Sunday, trading begins at 5:00 p.m.
Source: CME Interest Rate Product Guide and Calendar, 2007.
gets stolen, the insurance company pays some amount that depends on the car value, but
if the car does not get stolen, you receive nothing. Similarly, if you purchase an interest
rate option to cover yourself against an increase in future interest rates, you pay an up
front option premium. If the interest rate indeed increases, you receive a cash amount that
depends on the level of the reference interest rate, but if the interest rate does not increase,
you receive nothing. It is important to remember that you do not receive anything in the
good event: If the insurance company does not pay you it is because nobody stole your car,
which is good. If you bought insurance against an interest rate hike and the insurance did
not pay, this is good, as it means that the bad event did not occur.7
In a theft insurance policy, the amount of coverage and deductibles affect the level of
the insurance premium. Similarly, in financial options the amount of coverage depends on
the strike price: If you purchase an interest rate option to cover yourself against an increase
in interest rates, the higher the strike price and the lower is the payment if the interest rate
increases. This corresponds to lower insurance coverage, ex ante, and indeed it generates
a lower option premium that you have to pay up front. Finally, in a theft insurance policy,
the premium is higher when the risk that the car will be stolen is higher. Similarly, as we
7 Ex
post, of course, you wish you did not buy the insurance in this case, as you “wasted” the insurance (option)
premium. But this is an ex-post reasoning, while the hedging program has to be set up ex-ante, before knowing
what would happen.
218
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
are going to see, the interest rate option premium also depends on some type of interest
rate risk, its volatility.8
It is convenient to illustrate the idea of insurance in the hedging example discussed in
Examples 5.6 and 5.7 in Chapter 5.
EXAMPLE 6.2
Recall that today is March 1, 2001, and a firm is worried that the 6-month interest rate
could decline in the next six months, when it will receive a $100 million receivable.
The firm intends to invest this cash at T1 = 0.5 for the next six months by purchasing
safe Treasury bills. Let Pbill (T1 , T2 ) denote the six month T-bill at T1 = 0.5 with
maturity T2 = 1. In Example 5.6 in Chapter 5 the firm hedged by entering into a
forward contract with a bank. This forward contract, an OTC instrument, allowed the
firm to purchase six months later (T1 = 0.5) $100 million-worth of 6-month Treasury
bills for a price P f w d = $97.938, for $100 par value, specified today. The payoff of
the forward contract is then
Payoff of forward contract = Pbill (T1 , T2 ) − P f w d
That is, if the price of T-bills increases, which is what the firm worries about, then
the forward contract pays a positive amount.
Of course, if instead the interest rate increases, the forward contract generates
a negative cash flow, as the price of the T-bill Pbill (T1 , T2 ) would fall below the
forward price P f w d . A hedger may prefer to hedge only against the decrease in the
interest rate, but not against an increase. That is, she would prefer to purchase an
insurance against an interest rate decline. With this insurance, the firm can eliminate
the bad event (a lower interest rate) but keep the upside potential, that is, the fact that
if the interest rate instead increases, the firm can invest the $100 million receivable
to purchase T-bills at a lower price than P f w d .
A call option on the 6-month T-bill allows a firm to achieve exactly this latter
outcome. The call option enables the firm to purchase T-bills at maturity T1 for at
most the predetermined strike price, such as K = $97.938. This implies the payoff
at T1 is
Payoff of call option contract = max (Pbill (T1 , T2 ) − K, 0)
Of course, this payoff does not come for free and the firm has to pay the option
premium, which can be relatively expensive.
Assume that the bank asks the firm an option premium equal to9
Call(K) = $0.2701 (for $100 principal of T-bills),
for an option with maturity T1 = 0.5 and strike price K = $97.938.
8 There
are also many elements that are different between financial options and regular theft insurance. For
instance, deductibles are used in insurance market to attenuate the moral hazard problem, according to which full
coverage tend to induce a riskier behavior of the insured. An important difference between financial insurance
and theft insurance concern the calculation of the insurance (option) premium, as it is discussed in later chapters.
9 This premium is computed using the Ho-Lee model option pricing formula, developed in Chapter 17.
219
OPTIONS
How many options does the firm have to buy?
The total premium to be paid can be computed as follows: If the option will be
exercised, the firm wants to purchase $100 million-worth of T-bills at T1 at the strike
price K = $97.938. Therefore, the firm needs to purchase M = $100m/$97.938 =
1.02105 million of T-bills (with $100 principal). Assuming each option is defined
on a T-bill with $100 principal, then the firm needs to purchase M options. The total
cost of the option position is then
Total option premium = M × Call(K) = $275, 762.5
(6.13)
To summarize, by paying $275,762.5, the firm can insure against a decrease in the
interest rate. If the interest rate does not fall and perhaps in fact rises, then the
Pbill (T1 , T2 ) decreases, and it is better for the firm to purchase the 6-month T-bill in
the market, rather than exercise the option. If instead the interest rate does in fact
decrease, the firm can exercise the option and purchase the T-bill at K = $97.938
instead of at its higher market price, thereby hedging the bad outcome.
The previous example illustrates the calculation of the option’s premium to hedge against
a decline in interest rates. The next example follows up with the ex-post outcome and the
performance of the hedging strategy.
EXAMPLE 6.3
Consider Example 6.2. What happened next? The 6-month interest rate did in fact
decline, and the 6-month T-bill price on September 4, 2001, was Pbill (T1 , T2 ) =
$98.89 > $97.938 = K. In this case, the firm would exercise the option, and obtain
the payoff
Option payoff on September 4, 2001
= M × ($98.89 − $97.938)
=
$972, 043.54
This amount has to be invested by the firm together with the receivable of $100
million. The return on the additional amount makes up for the lower interest rate,
that is, for the higher price of the T-bill Pbill (T1 , T2 ) = $98.89. In fact, the total
investment in T-bills is given by
Investment in T-bills (with $100 principal)
=
=
$100, 000, 000 + $972, 043.54
$98.89
1, 021, 054.136
Because each T-bill yields $100 at T2 , the investment’s final payoff at T2 is $102,105,413.6.
In turn, this final payoff entails:
Final payoff at T2
1
−1
Gross annualized return on investment =
T2 − T1 Investment at T1
102, 105, 413.6
= 2×
−1
100, 000, 000
= 4.21%
(6.14)
220
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
This is the same rate of interest that the firm would have received with a regular
forward contract, as discussed in Example 5.7 in Chapter 5.
Gross versus Net Return. The calculation in Equation 6.14 is misleading about
the real rate of return realized by the option strategy. In fact, we haven’t taken into
account the option premium paid by the firm at time 0. To compute the fair return on
the option strategy and compare it to a forward contract, we must subtract the option
premium paid at time 0, given in Equation 6.13, from the final option payoff (Equation
6.14). To take into account the time value of money from 0 to T1 , we can use the
discount factor implied by the 6-month T-bill at time 0, Pbill (0, T1 ) = $97.728, i.e.
Z(0, T1 ) = 0.97728. We then obtain that the net option payoff on September 4, 2001
is
Net option payoff on September 4, 2001
=
$972, 043.54 −
=
$689, 870.06
$275, 762.5
0.97728
Using the net option payoff instead of the gross payoff in Equation 6.14, we find that
the investment in T-bills is given by
Investment in T-bills (with $100 principal)
$100, 000, 000 + $689, 870.06
$98.89
= 1, 018, 200.729
=
Given the implied final payoff at T2 is then equal to $101, 820, 072.9, the annualized
return on the $100 million receivable is
101, 820, 872.9
−1
Net annualized return on investment = 2 ×
100, 000, 000
= 3.64%
(6.15)
The lower rate obtained under the option contract is due to the premium paid to keep
the upside potential of a higher interest rate, a fact that that did not happen ex-post.
6.2.2
Option Strategies
Call and put options are the building blocks of more complex strategies. We can follow up
on Example 6.2 to illustrate some of the most common strategies.
6.2.2.1 Deductibles The firm in Example 6.2 may be interested in insuring only
against substantial declines in the 6-month interest rate, while keeping the risk for smaller
declines. In this case, it can purchase T-bill options with a higher strike price, which pay
only if the T-bill price increases substantially. This strategy would be cheaper, as the firm
retains some of the interest rate risk. For instance, setting a strike price K = $98.28 the
total cost of the option would be $137, 994.5, but the option would start paying off only if
the 6-month rate declines to rK = 2 × (1/K − 1) = 3.5%. Ex post, given that the T-bill
rate indeed moved to Pbill (T1 , T2 ) = $98.89 > 98.28, this strategy would have generated
an implied net rate of 3.215%, less than the rate computed under full coverage in Equation
6.15, as it is to be expected: The firm wanted to keep some risk so as to pay less insurance,
and when the bad outcome materializes it obtained a lower overall payoff.
OPTIONS
221
6.2.2.2 Collars Alternatively, the firm may decide to give up some of the return
stemming from an increase in interest rates in order to help pay for the premium to cover
against lower interest rates. In this case, it can sell some put option on T-bills, whose
premium can be used to offset the cost of the purchase of the call option described above.
This strategy is called a collar strategy. An especially popular collar strategy is one for
which the premium received from the put sold exactly offsets the premium for the call
purchased. That is, there is no ex-ante cash outflow from the collar strategy.
For instance, the firm in Example 6.2 could purchase call options with strike price
KC = $98.28 and sell put options with strike price KP = $97.596. Given the $100
million receivable, the firm needs to purchase exactly MC = $100 m/KC = 1, 017, 501
options with strike price KC and sell MP = 1, 024, 633 options with strike price KP .10
Using an option pricing formula that we develop in later chapters, the total cost of purchasing
the option equals the total gain from selling put. That is, the following relation holds
MC × Call(KC ) = MP × Call(KP )
If at T1 the interest rate declines, so that the T-bill price Pbill (T1 , T2 ) > KC = $98.28,
then the firm receives additional funding to compensate for the higher price of T-bills. This
is the same situation as in Example 6.2. In contrast, if the interest rate increases, so that the
T-bill price Pbill (T1 , T2 ) < KP = $97.596, then the firm will have to pay the put payoff
to the counterparty. This drain in cash will decrease the amount of investment that the firm
can make for the following six months. However, the drain is exactly compensated for by
the lower price of the T-bills (or higher interest rate), which the firm needs to purchase
for the next six months. Panel A of Figure 6.4 shows the profit from the collar strategy
for various values of the T-bill price at T1 . The collar pays when the T-bill price is high,
thereby hedging the lower interest rates, and requires a payment when the T-bill price is
low, compensated for though by the now higher rate.
Panel B of Figure 6.4 plots the implied rate of return from T1 to T2 that is obtained from
the hedging strategy for various scenarios at T1 . For easier reference, the x−axis now plots
the 6-month interest rate that is equivalent to the T-bill price on the x−axis in Panel A. For
this reason, the interest rate runs from a high number (low T-bill price in Panel A) to a low
number (high T-bill price in Panel A). The y−axis reports instead the implied interest rate,
computed exactly as in Equation 6.15, but for the collar strategy. We see that when the
interest rate declines (right-hand side of the plot), the effective rate realized by the collar
strategy also declines, but only down to 3.5%. If the interest rate declines further, the firm
is hedged. Similarly, if the interest rate increases (left-hand side of the plot), then the rate
of return for the firm increases, and the upside potential is realized. If the interest rate
increase above 4.93%(= 2 × (100/KP − 1) ), however, then the realized return flattens
out at that level. Any further increase in the interest rate would not generate any additional
return for the firm.
For comparison with the collar strategy, Figure 6.4 also plots the forward rate that the
firm would obtain by simply entering into a forward contract on the T-bill with a forward
price P f w d = $97.938, as already discussed in Example 5.7 in Chapter 5. The zero cost
collar strategy and the forward contract indeed share a common feature: They do not cost
anything at initiation. Therefore, the comparison of the effective rate at time T1 in Figure
10 These
KP .
numbers of options guarantee that the firm invests exactly $100 million-worth of T-bills at K C and at
222
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
Figure 6.4 Zero Cost Collar Strategy
6
1
A. Zero Cost Collar Profit
x 10
Profit
0.5
0
−0.5
−1
97
97.2
97.4
97.6
97.8
98
98.2
T−bill Price at T1
98.4
98.6
98.8
99
B. Implied Rate from Collar Strategy
5
Implied Rate (%)
Collar Strategy
Forward Contract
4.5
4
3.5
forward rate
at t=0
0.06
0.055
0.05
0.045
0.04
T−bill Rate at T1
0.035
0.03
0.025
6.4 provides an intuition of the tradeoff implied by collar strategy: By entering a long call
option and a short put option in a way to make the strategy zero cost, the firm is trading
off some upside (left-hand side of the figure) for some downside (right-hand side of the
figure). By choosing different strike prices the firm can tailor how much upside it wants to
trade for a better coverage on the downside.
6.2.2.3 Yield-Enhancing Strategies One popular set of strategies among investors
and investment banks goes under the name of yield enhancing strategies. The idea is the
following. Consider a coupon bond sold at par. The coupon rate equals, by definition, its
yield to maturity. To make the coupon higher, however, it is common to augment the bond
with a short option position. For instance, one could issue a bond that pays a fixed coupon
so long as a reference rate is below a given cutoff level, but such that the coupon would
decline if the interest rate rises above the cutoff. For instance, a possible strategy is to have
OPTIONS
223
a coupon defined as follows
Coupon at Ti = c − max(r(Ti−1 ) − rK , 0)
In other words, so long as the reference rate is below the strike rate rK , the coupon is fixed
at c. However, if the reference rate increases above rK , then the coupon is reduced. As
we have seen for inverse floaters, this type of security yields a higher coupon than market
so long as the rate r(Ti−1 ) < rK , but it would fall in price very quickly if the interest rate
increases above rK . Why is the coupon rate higher than with regular bonds? The reason
is that the investor who buys a bond with this coupon is effectively purchasing a straight
fixed rate bond with coupon c and selling an option (in fact, a cap). The premium from the
short cap position is embedded in the coupon rate, so that the bond still prices at par. We
investigate the risk and return of some of these structures in later chapters.
6.2.3
Put-Call Parity
Consider a collar strategy (see Subsection 6.2.2.2) long one call with strike price KC and
short one put with strike price KP with maturity T1 . Let these options be European,
meaning that they can be exercised only at maturity T1 . Consider now the special case in
which the strike prices are the same, that is, KC = KP = K. In this case, the payoff from
the collar strategy is
Payoff long call / Short put
= max(P (T1 , T2 ) − K, 0) − max(K − P (T1 , T2 ), 0)
= P (T1 , T2 ) − K
This is the same payoff of a long forward contract, establishing the fact that
Call − Put = Long forward
(6.16)
This relation is called put-call parity. In particular, the value of a forward contract on
a bond P (T1 , T2 ) for given delivery price K is given by
V f w d (0) = Z(0, T1 ) × P f w d (0, T1 , T2 ) − K
where P f w d (0, T1 , T2 ) is the forward price of the underlying bond P (T1 , T2 ). From the
put-call parity, we then have
Call(K) = P ut(K) + Z(0, T1 ) × P f w d (0, T1 , T2 ) − K
If we have put prices available, then we can compute call prices, and vice versa.
6.2.4
Hedging with Futures or with Options?
Once we consider options as insurance contracts, it is easier to think about the pros and
cons of options versus futures to hedge interest rate risk. The main trade off is costs versus
upside potential: Linear contracts such as futures, forwards, and swaps have no costs, but
also no upside. Options, on the other hand, cost money, and eliminate only the adverse
events.
One issue to take into account in the choice of futures (or forwards, or swaps) and options
is the hedger’s anticipated reaction to the good events. In other words, while futures pay
224
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
when the adverse event occurs, they also generate an outflow of cash from the firm when the
“good” event occurs. This means that corporate earnings, for instance, may be depressed
if the hedge loses money, notwithstanding the good outcome.11 Is the hedger willing to
accept the possibility of large cash outflows during good times? Would the CEO of the
company understand that the hedge is supposed to require money when the adverse event
is not occurring? When the trader in charge of the hedging program asks the CEO for the
cash amount in order to cover the futures’ margin calls, what would be the reaction of the
CEO? Would the trader be tagged as a rogue trader?
For instance, in Chapter 5, Example 5.9, a firm is engaged in a complicated hedging
strategy in order to match the cash flows of its liabilities, linked to a floating rate, to the
cash flows of its assets, which is a fixed rate receivable. The concern of the firm is that if the
LIBOR rate increases in the future, the cash outflows to pay for the debt may be larger than
the cash inflows from the receivables, generating a cash shortfall. The fixed-for-floating
swap discussed in Example 5.9 exactly matches the cash inflows with outflows, and the firm
is perfectly hedged going forward. After initiation of the contract, in March 2001, the 6month LIBOR did decrease substantially, and the sequence of realized LIBOR is reported
in Table 5.4. The last column of this table shows that the firm must make substantial
payments to honor the swap contract, even at the rate of many millions of dollars every six
months. Taken in isolation, and with the benefit of hindsight, the hedge with swaps was
not a good deal, as it looked like a bet on a future increase in interest rates, a bet gone
bad. It is very important, in these cases, to keep the focus on the overall strategy, that
is, the fact that because of the swap position, now the total net cash flow of the firm is
in fact zero, independent of the movement of interest rates. This was the objective of the
hedging strategy. Yet, if it is possible to forecast ex ante that large cash outflows generated
by the swap itself may generate internal problems between the trader and the CEO, for
instance, or the possibility of loss in competitiveness – other competing firms may have
lower borrowing rates if the interest rate declines – then an insurance contract may be
preferable. Indeed, the insurance contract can be tailored to eliminate only the really bad
outcomes, that may hamper the firm from functioning.
EXAMPLE 6.4
Consider again Example 5.9 in Chapter 5. An insurance strategy may be as follows:
At the moment, the 6-month LIBOR rate r2 (0) = 4.95% is sufficiently low, to
ensure that the semi-annual receivable of $5.5 million covers the floating coupon
due c(0.5) = (r2 (0) + 4bps) × 0.5 × 200 m = 4.99 million. However, were the
LIBOR rate to increase above 5.46%, then the receivable of $5.5 million would not be
sufficient to pay for the floating rate coupon. An insurance is to purchase a sequence
of interest rate options that pay anytime the 6-month LIBOR is above rK = 5.46%.
This option is called a cap. In particular, a five year, semi-annual cap pays a cash
flow at every Ti equal to
Cap cash flow at Ti = N × Δ × max (r2 (Ti−1 ) − rK , 0)
11 This
problem is partly mitigated by hedge accounting, which allows a firm to match these negative cash flows
against the (positive) value of the primary risk exposure being hedged. However, hedge accounting not always is
feasible.
SUMMARY
225
where Δ = Ti − Ti−1 is the time between payments, N is the notional amount, and
r2 (t) is the 6-month LIBOR rate at t. Like the floating rate bond issued by the firm,
the cap pays at time Ti an amount linked to the LIBOR defined at the previous reset
date, Ti−1 .
Given a notional amount N = 200 million, the premium for such a cap is
5-year cap premium = $5, 998, 733
By paying this amount, the firm covers itself from hikes in the interest rate for the
next five years, but it would not have to make any payment in case the interest rate
declines. That is, it insured itself against increases in interest rates.
As we know from Table 5.4 in Chapter 5, the 6-month LIBOR indeed declined
during the following five years. In this case, the option would expire worthless, as it
would make no payment. Although there is a temptation to think that the firm lost
money on the option, as the firm paid almost $6 million for it, we must recall that the
decline in interest rates is a good outcome for the firm, as now the firm has to make
lower payments from the liabilities, but still obtain a fixed $5.5 million receivable.
The cap was an insurance policy.
As mentioned earlier, the option premium can be decreased by following some
other strategies. For instance, if the firm is willing to bear some more interest rate
risk (a deductible in the insurance terminology), it could insure itself only for interest
rate increases above rK = 6.5% instead of rK = 5.46%. In this case, the option
premium would drop to approximately $3.4 million. Similarly, the firm may decide
to keep the upside potential only within a given interval of interest rates, and therefore
buy a collar, meaning that it can buy the cap to protect against interest rates, but also
sell a floor so as to partially pay for it.
6.3
SUMMARY
In this chapter we covered the following topics:
1. Futures: A futures contract is similar to a forward contract, in which the counterparty
short the contract agrees to sell a prespecified security on a prespecified date and at a
prespecified price to the counterparty long the contract. The latter agrees to buy the
security and to pay the prespecified price. Some futures contracts are cash settled,
meaning that no exchange of security actually takes place. Characteristics of futures
contracts are that they are:
• Traded in regulated exchanges.
• Standardized: The maturity of the contracts as well as the delivery securities
are decided by the exchange.
• Marked-to-market: Profits and losses accrue to the counterparties on a daily
basis.
2. Convergence property: This is when the futures price converges at maturity to the
price of the unlderlying contract.
226
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
3. Options: Options are contracts in which the counterparty long the contract has the
right, but not the obligation, to purchase (for a call) or sell (for a put) the security
underlying the contract, at a prespecified price and within a prespecified time. The
counterparty short the contract is obliged to sell (if a call) or purchase (if a put) the
security at the prespecified price.
4. European and American Options: European options are options that can be exercised
only at maturity. American options are options that can be exercised anytime before
maturity.
5. Put-Call parity: For European options, a put minus a call with the same strike price
equals a forward contract. Therefore, a put equals a call minus a forward contract.
6. Collar Strategies: An options’ portfolio that is long a call and short a put option.
6.4 EXERCISES
1. This exercise uses the data in Table 6.9. Suppose that on February 15, 1994 a firm
wants to enter into a forward contract to purchase 5-year Treasuries, with coupon
rate 6%, in two years:
(a) Compute the forward price.
(b) What is the value of the forward contract at initiation?
(c) Compute the value of the forward contract on each of the next five days.
(d) Assume the firm and the counterparty mark to market the forward contract;
describe the cash flows between the counterparties over time.
(e) Panel B contains overnight rates. Compute the total profit / loss on the contract
after two, three, four, and five days.
2. On March 21, 2007 a firm enters into 100 90-day Eurodollar futures contracts
(contract size is $1,000,000). The quoted futures price on this day is: $93.695. Over
the life of the contracts, prices move as in Table 6.10.
(a) What will the daily P&L from futures for the quoted dates be?
(b) What will the cumulative P&L for the quoted dates be?
(c) You find out that 90-day Eurodollar futures are subjected to the following
requirements: Initial Margin: $1,485 (per contract); Maintenance Margin:
$1,110 (per contract).
i. What will the cash flow of the contract be, assuming that the firm never
takes money from the margin account?
ii. What will the cash flow of the contract be if the firm decides to take every
profit from the contract instead of leaving it in the margin account?
3. Consider the data in Table 6.11, where P f u t (t, T1 ) is the price of a 90-day Eurodollar
futures contract expiring on April 14, 2008; f (t, T1 , T2 ) is the time t forward rate
EXERCISES
Table 6.9
227
Semi-annually Compounded Yield Curves
Date
2/15/1994
2/16/1994
5/15/1994
8/15/1994
11/15/1994
2/15/1995
5/15/1995
8/15/1995
11/15/1995
2/15/1996
5/15/1996
8/15/1996
11/15/1996
2/15/1997
5/15/1997
8/15/1997
11/15/1997
2/15/1998
5/15/1998
8/15/1998
11/15/1998
2/15/1999
5/15/1999
8/15/1999
11/15/1999
2/15/2000
5/15/2000
8/15/2000
11/15/2000
2/15/2001
5/15/2001
8/15/2001
11/15/2001
3.57%
3.59%
3.80%
3.86%
4.01%
4.18%
4.27%
4.48%
4.58%
4.62%
4.76%
4.81%
4.95%
5.04%
5.14%
5.20%
5.25%
5.33%
5.38%
5.45%
5.49%
5.50%
5.57%
5.61%
5.65%
5.71%
5.76%
5.78%
5.83%
5.85%
5.92%
3.69%
3.59%
3.84%
3.89%
4.03%
4.18%
4.29%
4.48%
4.58%
4.62%
4.77%
4.81%
4.96%
5.04%
5.13%
5.19%
5.25%
5.34%
5.38%
5.45%
5.50%
5.50%
5.56%
5.61%
5.66%
5.72%
5.77%
5.81%
5.84%
5.88%
5.94%
2/16/1994
3.54%
Panel A: Yeld Curves
2/17/1994 2/18/1994 2/22/1994
3.81%
3.71%
3.70%
3.92%
4.05%
4.24%
4.36%
4.56%
4.65%
4.70%
4.86%
4.89%
5.04%
5.13%
5.22%
5.30%
5.36%
5.45%
5.50%
5.57%
5.61%
5.62%
5.68%
5.71%
5.76%
5.82%
5.87%
5.92%
5.95%
5.96%
6.03%
4.32%
3.55%
3.72%
4.08%
4.12%
4.32%
4.43%
4.55%
4.71%
4.75%
4.90%
4.94%
5.11%
5.19%
5.29%
5.37%
5.43%
5.53%
5.56%
5.63%
5.68%
5.68%
5.76%
5.80%
5.85%
5.91%
6.01%
5.99%
6.03%
6.07%
6.13%
3.15%
3.49%
3.66%
4.07%
4.21%
4.34%
4.46%
4.65%
4.73%
4.73%
4.90%
4.92%
5.09%
5.19%
5.29%
5.36%
5.41%
5.50%
5.54%
5.61%
5.66%
5.66%
5.73%
5.80%
5.84%
5.90%
5.94%
5.96%
6.03%
6.08%
6.13%
Panel B: Overnight Rates
2/17/1994 2/18/1994 2/22/1994
3.78%
3.90%
4.44%
2/23/1994
3.15%
3.54%
3.71%
4.19%
4.36%
4.44%
4.59%
4.75%
4.83%
4.81%
5.00%
5.01%
5.18%
5.27%
5.38%
5.41%
5.50%
5.58%
5.62%
5.70%
5.74%
5.75%
5.82%
5.87%
5.91%
5.93%
6.03%
6.04%
6.11%
6.15%
6.21%
2/23/1994
2.88%
c 2009 Center for Research
Notes: Yields calculated based on data from CRSP (Daily Treasuries) in Security Prices (CRSP), The University of Chicago Booth School of Business.
228
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
Table 6.10 Eurdollar Futures
Date
Price
Date
Price
Date
Price
Date
Price
21-Mar-07
22-Mar-07
23-Mar-07
26-Mar-07
27-Mar-07
28-Mar-07
29-Mar-07
30-Mar-07
2-Apr-07
3-Apr-07
4-Apr-07
5-Apr-07
6-Apr-07
9-Apr-07
10-Apr-07
11-Apr-07
12-Apr-07
16-Apr-07
17-Apr-07
18-Apr-07
19-Apr-07
20-Apr-07
23-Apr-07
24-Apr-07
25-Apr-07
26-Apr-07
27-Apr-07
30-Apr-07
1-May-07
2-May-07
3-May-07
4-May-07
93.635
93.685
93.570
93.500
93.320
93.330
93.330
93.375
93.340
93.355
93.380
93.340
93.455
93.395
93.245
93.240
93.175
93.060
93.170
93.265
93.080
93.060
93.115
93.100
93.020
93.135
93.015
93.025
93.085
93.135
93.250
93.250
7-May-07
8-May-07
9-May-07
10-May-07
11-May-07
14-May-07
15-May-07
16-May-07
17-May-07
18-May-07
21-May-07
22-May-07
23-May-07
24-May-07
25-May-07
29-May-07
30-May-07
31-May-07
1-Jun-07
4-Jun-07
5-Jun-07
6-Jun-07
7-Jun-07
8-Jun-07
11-Jun-07
12-Jun-07
13-Jun-07
14-Jun-07
15-Jun-07
18-Jun-07
19-Jun-07
20-Jun-07
93.185
93.175
93.245
93.095
92.825
92.935
92.870
92.895
93.010
93.090
93.120
93.120
93.050
92.925
92.915
92.895
92.865
93.010
93.060
93.120
93.195
93.225
93.140
93.065
93.140
93.185
93.150
93.145
93.020
92.995
93.010
93.060
21-Jun-07
22-Jun-07
25-Jun-07
26-Jun-07
27-Jun-07
28-Jun-07
29-Jun-07
2-Jul-07
3-Jul-07
5-Jul-07
6-Jul-07
9-Jul-07
10-Jul-07
11-Jul-07
12-Jul-07
13-Jul-07
16-Jul-07
17-Jul-07
18-Jul-07
19-Jul-07
20-Jul-07
23-Jul-07
24-Jul-07
25-Jul-07
26-Jul-07
27-Jul-07
30-Jul-07
31-Jul-07
1-Aug-07
2-Aug-07
3-Aug-07
6-Aug-07
93.095
93.210
93.190
93.080
93.090
93.015
92.945
92.965
92.975
92.930
92.950
92.995
93.035
93.025
93.065
93.120
93.185
93.185
93.265
93.265
93.260
93.295
93.290
93.190
93.170
93.235
93.230
93.310
93.265
93.190
93.140
93.160
7-Aug-07
8-Aug-07
9-Aug-07
10-Aug-07
13-Aug-07
14-Aug-07
15-Aug-07
16-Aug-07
17-Aug-07
20-Aug-07
21-Aug-07
22-Aug-07
23-Aug-07
24-Aug-07
27-Aug-07
28-Aug-07
29-Aug-07
30-Aug-07
31-Aug-07
4-Sep-07
5-Sep-07
6-Sep-07
7-Sep-07
10-Sep-07
11-Sep-07
13-Sep-07
14-Sep-07
17-Sep-07
18-Sep-07
19-Sep-07
20-Sep-07
21-Sep-07
93.150
93.260
93.205
93.250
93.340
93.335
93.335
93.370
93.445
93.380
93.415
93.405
93.440
93.390
93.385
93.480
93.620
93.565
93.510
93.345
93.370
93.510
93.515
93.450
93.505
93.490
93.620
93.540
93.420
93.485
93.495
93.590
Source: Bloomberg.
EXERCISES
Table 6.11
Date
16-Oct-07
17-Oct-07
18-Oct-07
19-Oct-07
22-Oct-07
23-Oct-07
24-Oct-07
25-Oct-07
26-Oct-07
29-Oct-07
30-Oct-07
31-Oct-07
1-Nov-07
2-Nov-07
5-Nov-07
6-Nov-07
7-Nov-07
8-Nov-07
9-Nov-07
12-Nov-07
13-Nov-07
14-Nov-07
15-Nov-07
16-Nov-07
19-Nov-07
20-Nov-07
21-Nov-07
23-Nov-07
26-Nov-07
27-Nov-07
28-Nov-07
29-Nov-07
30-Nov-07
3-Dec-07
4-Dec-07
5-Dec-07
6-Dec-07
7-Dec-07
10-Dec-07
11-Dec-07
12-Dec-07
13-Dec-07
14-Dec-07
17-Dec-07
18-Dec-07
19-Dec-07
20-Dec-07
21-Dec-07
24-Dec-07
27-Dec-07
28-Dec-07
31-Dec-07
2-Jan-08
3-Jan-08
4-Jan-08
7-Jan-08
8-Jan-08
9-Jan-08
10-Jan-08
11-Jan-08
14-Jan-08
Futures and Forward Rates
P f u t (t, T 1 ) f (t, T 1 , T 2 ) Z (t, T 1 ) Z (t, t + dt)
95.3900
95.5150
95.5700
95.6850
95.6700
95.6700
95.7750
95.7750
95.7350
95.7000
95.6800
95.5500
95.6700
95.7950
95.7250
95.7050
95.7550
95.8950
95.9300
95.9300
95.8000
95.7900
95.9450
95.9450
95.8350
95.8300
95.9500
95.9500
96.0950
95.9700
95.8900
95.9850
96.0000
95.9700
95.9650
95.9650
95.8750
95.7400
95.7250
95.7050
95.8100
95.7350
95.6400
95.7000
95.7850
95.8450
95.8700
95.8250
95.8050
95.8050
95.8850
95.9350
96.0600
96.0850
96.1950
96.2200
96.2300
96.3300
96.4000
96.5800
96.6300
Source: Bloomberg.
4.9377%
4.8740%
4.7691%
4.6475%
4.5410%
4.6208%
4.5184%
4.4484%
4.4708%
4.4837%
4.5514%
4.5478%
4.6967%
4.5323%
4.5183%
4.5836%
4.5498%
4.5025%
4.4121%
4.3647%
4.3561%
4.4676%
4.4416%
4.3667%
4.4307%
4.4359%
4.4164%
4.3945%
4.4663%
4.3768%
4.4886%
4.4769%
4.4795%
4.4383%
4.4606%
4.4754%
4.4785%
4.5479%
4.6552%
4.6395%
4.6333%
4.5007%
4.5878%
4.6172%
4.5908%
4.5211%
4.4680%
4.4625%
4.4741%
4.5054%
4.4921%
4.4191%
4.3892%
4.2369%
4.2555%
4.1396%
4.1183%
4.0523%
3.9889%
3.8858%
3.6526%
0.9751
0.9753
0.9756
0.9761
0.9768
0.9769
0.9772
0.9776
0.9777
0.9781
0.9784
0.9784
0.9784
0.9787
0.9790
0.9790
0.9792
0.9794
0.9797
0.9801
0.9803
0.9803
0.9803
0.9804
0.9806
0.9807
0.9808
0.9810
0.9813
0.9815
0.9815
0.9816
0.9817
0.9821
0.9822
0.9823
0.9825
0.9826
0.9829
0.9831
0.9833
0.9836
0.9838
0.9842
0.9844
0.9846
0.9849
0.9851
0.9855
0.9859
0.9863
0.9866
0.9869
0.9872
0.9874
0.9879
0.9882
0.9884
0.9887
0.9891
0.9900
229
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
0.9998
Date
15-Jan-08
16-Jan-08
17-Jan-08
18-Jan-08
22-Jan-08
23-Jan-08
24-Jan-08
25-Jan-08
28-Jan-08
29-Jan-08
30-Jan-08
31-Jan-08
1-Feb-08
4-Feb-08
5-Feb-08
6-Feb-08
7-Feb-08
8-Feb-08
11-Feb-08
12-Feb-08
13-Feb-08
14-Feb-08
15-Feb-08
19-Feb-08
20-Feb-08
21-Feb-08
22-Feb-08
25-Feb-08
26-Feb-08
27-Feb-08
28-Feb-08
29-Feb-08
3-Mar-08
4-Mar-08
5-Mar-08
6-Mar-08
7-Mar-08
10-Mar-08
11-Mar-08
12-Mar-08
13-Mar-08
14-Mar-08
17-Mar-08
18-Mar-08
19-Mar-08
20-Mar-08
25-Mar-08
26-Mar-08
27-Mar-08
28-Mar-08
31-Mar-08
1-Apr-08
2-Apr-08
3-Apr-08
4-Apr-08
7-Apr-08
8-Apr-08
9-Apr-08
10-Apr-08
11-Apr-08
14-Apr-08
P f u t (t, T 1 ) f (t, T 1 , T 2 ) Z (t, T 1 ) Z (t, t + dt)
96.5900
96.5700
96.6250
96.7150
97.1500
97.2450
97.0350
97.0650
97.1250
97.1050
97.1850
97.2400
97.1700
97.1450
97.2400
97.2700
97.2650
97.2900
97.2950
97.2900
97.2800
97.2650
97.2600
97.1900
97.1550
97.2000
97.1500
97.1200
97.1450
97.1900
97.2500
97.3800
97.3550
97.3400
97.3000
97.3350
97.4800
97.4850
97.4000
97.4450
97.4900
97.6350
97.7700
97.5500
97.5600
97.5275
97.4350
97.4250
97.4275
97.4250
97.4400
97.3800
97.3200
97.3075
97.3600
97.3150
97.2550
97.2800
97.2800
97.2850
97.2912
3.6628%
3.6452%
3.7057%
3.6295%
3.3133%
3.0388%
3.0847%
3.2957%
3.1394%
3.1414%
3.1548%
3.0024%
2.9729%
3.0675%
3.0789%
2.9475%
2.9297%
2.9512%
2.9227%
2.9167%
2.9124%
2.9325%
2.9446%
2.9701%
3.0113%
3.0612%
3.0026%
3.0499%
3.0541%
3.0192%
2.9751%
2.9505%
2.8894%
2.8963%
2.8999%
2.9099%
2.8230%
2.8266%
2.7963%
2.8098%
2.7437%
2.7150%
2.4805%
2.4785%
2.5706%
2.5822%
2.6602%
2.6671%
2.6716%
2.6844%
2.6704%
2.6633%
2.6937%
2.7304%
2.7352%
2.7120%
2.7066%
2.7082%
2.6974%
2.6974%
2.7088%
0.9902
0.9904
0.9906
0.9908
0.9916
0.9926
0.9928
0.9928
0.9932
0.9933
0.9934
0.9936
0.9938
0.9939
0.9940
0.9941
0.9942
0.9943
0.9946
0.9947
0.9948
0.9949
0.9950
0.9953
0.9954
0.9955
0.9956
0.9958
0.9959
0.9960
0.9961
0.9963
0.9965
0.9966
0.9967
0.9968
0.9969
0.9972
0.9974
0.9975
0.9976
0.9977
0.9981
0.9982
0.9982
0.9983
0.9986
0.9987
0.9987
0.9988
0.9990
0.9990
0.9991
0.9992
0.9993
0.9995
0.9995
0.9996
0.9997
0.9997
1.0000
0.9998
0.9998
0.9998
0.9998
0.9998
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9998
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
0.9999
230
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
for 90-day LIBOR at April 14, 2008; Z(t, T1 ) is the LIBOR discount from t to April
14, 2008; and Z(t, t + dt) is the overnight LIBOR discount (where dt = 1 day).
(a) On April 14, 2008 the 90-day LIBOR was 2.7088%. Did the rates converge?
(b) What was the total profit / loss on April 14, 2008 of a futures contract entered
on October 16, 2007?
(c) What would be the total profit / loss on April 14, 2008 of a forward contract
entered on October 16, 2007?
(d) Calculate the total profit / loss at each date for the futures contract.
(e) Calculate the total profit / loss at each date for the forward contract.
(f) How close are profits / losses over time? Plot the values.
4. Suppose that on October 16, 2007 a firm loses a lawsuit and has to pay $100 million
within 9 months. Currently it has no money to pay, so it decides to sell a piece of
equipment. Although the equipment could go for up to $150 million, the firm is
desperate and is willing to take much less so as to make the lawsuit payment. A
buyer comes up, who offers to buy the equipment at a significant discount and will
pay for it in six months. The firm’s CFO thinks that the deal might work if they enter
into a futures contract or a forward contract.
(a) Using only the information available on October 16, 2007 in Table 6.11, what is
the least that the firm should accept, if it decides to enter into a futures contract?
(b) Using only the information available on October 16, 2007 what is the least that
the firm should accept, if it decides to enter into a forward contract?
5. Following up on Exercise 4, let it now be April 16, 2008. The 90-day LIBOR rate is
2.7088%. Using the information in Exercise 4:
(a) What is the total amount available to invest in the 90-day LIBOR from the
futures contract?
(b) What is the total amount available to invest in the 90-day LIBOR from the
forward contract?
(c) After 90 days, does the firm receive enough money from the futures contract?
Is it exactly the amount that is needed?
(d) After 90 days, does the firm receive enough money from the forward contract?
Is it exactly the amount that is needed?
6. Consider again Exercise 4. Imagine that back on October 16, 2007, the firm decided
to sell the equipment for $98.78 million receiving the cash in 6 months. The CFO
looks at the 3-month LIBOR at the time and sees that it is at 5.2088%. If the rate
holds for only 6 months he could invest the proceeds from the sale and make a
$67,000 gain because the return will be $100.067 million at the time of the lawsuit
payment. So, he decides to forgo any type of hedging.
(a) Given past data on the 3-month LIBOR, compute the statistical distribution of
possible cash flows in nine months. Specifically, proceed as follows:
EXERCISES
231
i. Using past data, compute the change in the 3-month LIBOR over six-month
periods. Use these changes as possible scenarios about the fluctuations in
the 3-month LIBOR over the next six months, and compute the distribution
of the total amount available in nine months. How frequently would the
firm have enough cash to pay for the lawsuit?
ii. Altenatively, use past data to run the regression
rt+ 1 = α + βrt + t+1
where rt is the LIBOR rate at time t, and t+1 ∼ N (0, σ 2 ). Use the
estimated parameters α, β and σ together with the current LIBOR rate
rtoday = 5.2088% to simulate LIBOR rates at t = today+ six months.
According to this calculation, how frequently would the firm have enough
cash to pay for the lawsuit?
(b) On an ex post basis, will the firm have enough cash to pay the lawsuit? How
much will the surplus / deficit be?
7. On October 16, 2007, the firm from Exercise 4 decides to enter into a Eurodollar
futures contract with expiration on April 14, 2008, so it buys 100 contracts (each is
worth $1,000,000). The firm decides that it will create a separate account in which
it will simply store the daily P/L of the futures contracts until expiration.
(a) Using the information in Table 6.11, how much money will the firm receive in
six months?
(b) Suppose that futures prices moved instead according to Table 6.12. How much
money will the firm receive in six months?
8. In the previous exercises we assumed that the firm just sat on the profits / losses from
the futures contract. This is quite unrealistic because the firm would either invest the
profits or borrow to cover the losses from the contract. Assume that the firm decides
to do so by investing / borrowing from the time of the profit / loss until the expiration
of the contract. In this case we know from Table 6.11 the present value of a dollar on
T1 [see the column for Z(t, T1 )], if we want to know the value of receiving a dollar
now (t) at a future date T1 , all we have to do is divide the dollar by Z(t, T1 ).
(a) Compute the profit / loss of the firm for the futures prices presented in Exercise
3.
(b) What is the total return for the firm after investing the money?
(c) Assuming that the discounts Z(t, T1 ) in Table 6.11 also apply for the alternative
scenario for futures in Table 6.12, compute for this latter case the profit / loss
of the firm from the futures position.
(d) In the latter case, what is the total return for the firm after investing the money?
9. Comparing the results from Exercises 7 and 8 we can see that whether the firm
invests/borrows the proceeds from the hedging activity yields quite a different final
amount at maturity T1 . In particular, it appears that the firm is buying too many
futures contracts. The adjustment to this excess is called tailing the hedge. To do
232
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
Table 6.12
Alternative Futures Price Movement
Date
Price
Date
Price
Date
Price
Date
Price
16-Oct-07
17-Oct-07
18-Oct-07
19-Oct-07
22-Oct-07
23-Oct-07
24-Oct-07
25-Oct-07
26-Oct-07
29-Oct-07
30-Oct-07
31-Oct-07
1-Nov-07
2-Nov-07
5-Nov-07
6-Nov-07
7-Nov-07
8-Nov-07
9-Nov-07
12-Nov-07
13-Nov-07
14-Nov-07
15-Nov-07
16-Nov-07
19-Nov-07
20-Nov-07
21-Nov-07
23-Nov-07
26-Nov-07
27-Nov-07
28-Nov-07
95.390
95.265
95.210
95.095
95.110
95.110
95.005
95.005
95.045
95.080
95.100
95.230
95.110
94.985
95.055
95.075
95.025
94.885
94.850
94.850
94.980
94.990
94.835
94.835
94.945
94.950
94.830
94.830
94.685
94.810
94.890
29-Nov-07
30-Nov-07
3-Dec-07
4-Dec-07
5-Dec-07
6-Dec-07
7-Dec-07
10-Dec-07
11-Dec-07
12-Dec-07
13-Dec-07
14-Dec-07
17-Dec-07
18-Dec-07
19-Dec-07
20-Dec-07
21-Dec-07
24-Dec-07
27-Dec-07
28-Dec-07
31-Dec-07
2-Jan-08
3-Jan-08
4-Jan-08
7-Jan-08
8-Jan-08
9-Jan-08
10-Jan-08
11-Jan-08
14-Jan-08
15-Jan-08
94.795
94.780
94.810
94.815
94.815
94.905
95.040
95.055
95.075
94.970
95.045
95.140
95.080
94.995
94.935
94.910
94.955
94.975
94.975
94.895
94.845
94.720
94.695
94.585
94.560
94.550
94.450
94.380
94.200
94.150
94.190
16-Jan-08
17-Jan-08
18-Jan-08
22-Jan-08
23-Jan-08
24-Jan-08
25-Jan-08
28-Jan-08
29-Jan-08
30-Jan-08
31-Jan-08
1-Feb-08
4-Feb-08
5-Feb-08
6-Feb-08
7-Feb-08
8-Feb-08
11-Feb-08
12-Feb-08
13-Feb-08
14-Feb-08
15-Feb-08
19-Feb-08
20-Feb-08
21-Feb-08
22-Feb-08
25-Feb-08
26-Feb-08
27-Feb-08
28-Feb-08
94.210
94.155
94.065
93.630
93.535
93.745
93.715
93.655
93.675
93.595
93.540
93.610
93.635
93.540
93.510
93.515
93.490
93.485
93.490
93.500
93.515
93.520
93.590
93.625
93.580
93.630
93.660
93.635
93.590
93.530
29-Feb-08
3-Mar-08
4-Mar-08
5-Mar-08
6-Mar-08
7-Mar-08
10-Mar-08
11-Mar-08
12-Mar-08
13-Mar-08
14-Mar-08
17-Mar-08
18-Mar-08
19-Mar-08
20-Mar-08
25-Mar-08
26-Mar-08
27-Mar-08
28-Mar-08
31-Mar-08
1-Apr-08
2-Apr-08
3-Apr-08
4-Apr-08
7-Apr-08
8-Apr-08
9-Apr-08
10-Apr-08
11-Apr-08
14-Apr-08
93.4000
93.4250
93.4400
93.4800
93.4450
93.3000
93.2950
93.3800
93.3350
93.2900
93.1450
93.0100
93.2300
93.2200
93.2525
93.3450
93.3550
93.3525
93.3550
93.3400
93.4000
93.4600
93.4725
93.4200
93.4650
93.5250
93.5000
93.5000
93.4950
93.4888
APPENDIX: LIQUIDITY AND THE LIBOR CURVE
233
this we must find a ‘tailing factor’ through which we adjust the number of futures
contracts in order to return to the desired levels (the ones obtained from Exercise 7):
Tailed hedge = Untailed hedge × Tailing factor
Looking at the work done to calculate the future value of investing/borrowing the
profits/losses from the futures contract we see that the key difference is when we
divide by Z(t, T1 ). To undo this we need to multiply again by Z(t, T1 ). For our
specific case we have that in the untailed hedge we bought 100 contracts which means
that:
Tailed hedge = 100 × Z(t, T1 )
Note that the match is not exact because we compute the tailing factor at one period,
multiplying by Z(t, T1 ), and invest in the next, dividing by Z(t + 1, T1 ).
(a) After ‘tailing the hedge’, how much money does the firm receive in six months
if futures prices move according to Table 6.11? Is it different than the original
computation?
(b) After ‘tailing the hedge’, how much money does the firm receive in six months
if futures prices move according to Table 6.12? Is it different than the original
computation?
10. Today is t = 0. You are given the following data:
• The 6-month zero coupon bond is priced at $98.24
• The 9-month zero coupon bond is priced at $97.21
• Call option (European) on the 13 week Treasury bill with maturity in 6-months
and strike price of $99.12 is priced at $0.2934
• Put option (European) on the 13 week Treasury bill with maturity in 6-months
and strike price of $99.12 is priced at $0.1044
(a) Are the securities priced correctly?
(b) Assume that someone tells you that she is 100% sure that the call option is
priced correctly. Can you design a strategy to take advantage of the arbitrage
opportunity, if there is one?
6.5
APPENDIX: LIQUIDITY AND THE LIBOR CURVE
In this section we discuss in more detail some issues related to the construction of the LIBOR
curve, introduced in Section 5.4.3 in Chapter 5. The LIBOR curve is the reference discount
curve that traders use to discount LIBOR-based cash flows. One important complication
in the computation of the LIBOR curve is the fact that swaps, which are over-the-counter
securities, are not very liquid instruments. Therefore the LIBOR curve extracted from
swaps may not accurately reflect the true time value of money, but it contains a premium
for liquidity. Whenever possible, using liquid exchange-traded securities is preferable. For
the LIBOR curve, in particular, the use of Eurodollar futures is preferable to swaps, as
Eurodollar futures, described in Table 6.3, are among the most traded, and liquid, futures
234
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
Table 6.13
Interest Rate Data on May 5, 2008
LIBOR Rates
Maturity
Month
Years
1
0.0833
2
0.1667
3
0.2500
4
0.3333
5
0.4167
6
0.5000
12
1.0000
EURODOLLAR FUTURES
Rates (%)
2.6975
2.7375
2.7700
2.8044
2.8400
2.8737
2.9938
Swap Rates
Maturity
(Year)
1
2
3
4
5
7
10
Bid
2.857
3.191
3.484
3.718
3.891
4.174
4.448
Ask
2.865
3.194
3.49
3.721
3.896
4.177
4.453
Contract
EDM8
EDU8
EDZ8
EDH9
EDM9
EDU9
EDZ9
EDH0
EDM0
EDU0
EDZ0
EDH1
EDM1
EDU1
EDZ1
EDH2
EDM2
EDH2
EDZ2
Futures
Maturity
Jun08
Sep08
Dec08
Mar09
Jun09
Sep09
Dec09
Mar10
Jun10
Sep10
Dec10
Mar11
Jun11
Sep11
Dec11
Mar12
Jun12
Sep12
Dec12
Time to
Maturity
0.112
0.364
0.614
0.860
1.112
1.364
1.614
1.860
2.112
2.364
2.614
2.860
3.112
3.364
3.614
3.863
4.115
4.367
4.616
Quote
(Mid)
97.350
97.300
97.070
96.900
96.700
96.535
96.325
96.195
96.050
95.925
95.795
95.725
95.645
95.570
95.480
95.430
95.370
95.310
95.235
Futures
Rate (%)
2.650
2.700
2.930
3.100
3.300
3.465
3.675
3.805
3.950
4.075
4.205
4.275
4.355
4.430
4.520
4.570
4.630
4.690
4.765
Implied (c.c.)
Forward
Rate (%)
2.641
2.690
2.917
3.083
3.279
3.439
3.643
3.767
3.906
4.023
4.146
4.208
4.279
4.345
4.425
4.465
4.514
4.562
4.625
Source: Bloomberg.
in the world. Indeed, traders typically compute the LIBOR curve from three sources of
information: (1) LIBOR fixes for overnight or very short maturities, (2) Eurodollar futures
for maturities up to three years – liquidity is low for longer maturities – and, (3) swaps for
longer maturities.
Table 6.13 provides the data for this exercise, obtained on May 5, 2008. First, the
LIBOR rates in Column 3 provide the discounts for short maturities. For simplicity, we
extract the LIBOR curve at a quarterly frequency. Denoting by r4 (t) the 3-month LIBOR
rate at t, the first discount is then given by
Z(0, 0.25) =
1
1
=
= 99.3123%
1 + r4 (0) × 0.25
1 + .0277 × 0.25
Columns 4 to 7 report Eurodollar futures prices P f u t (0, Ti , Ti+1 ) where Ti+1 = Ti +
0.25. From the definition of the Eurodollar futures convention, we can convert these futures
prices into futures rates as f4f u t (0, Ti , Ti+ 1 ) = 100 − P (0, Ti , Ti+1 ), where the subscript
“4” denotes the quarterly compounding. Column “futures rates” in Table 6.13 reports the
outcome of this calculation.
The next question is how to use futures rates to obtain the discount curve. If these were
forward rates instead of futures rates, we could use the methodology illustrated in Section
5.1.3 in Chapter 5 to obtain the discount factors Z(0, T ). Luckily, we have seen in Section
6.1.4 that futures and forwards prices (and rates) are related to each other. Specifically, Fact
6.2 shows that under two particular assumptions, they are in fact the same. As discussed
right afterward, these assumptions are both violated, especially for Eurodollar futures.
However, as we shall see in Chapter 21, there is a simple correction that can be made to
convert futures rates into forward rates.
APPENDIX: LIQUIDITY AND THE LIBOR CURVE
235
Fact 6.3 The continuously compounded forward rate f (0, Ti , Ti+1 ) and futures rate
f f u t (0, Ti , Ti+1 ) are related by
1
f (0, Ti , Ti+ 1 ) = f f u t (0, Ti , Ti+1 ) − σ 2 Ti Ti+1
2
(6.17)
where σ is the annualized volatility of the underlying LIBOR rate.12
The volatility σ can be computed from the history of interest rate movements.13 Specifically, from historical data we can compute the annualized standard deviation of changes
in the 1-month LIBOR rate, which gives us σ = 0.01. The last column of Table 6.13
contains the continously compounded forward rates obtained from Equation 6.17. Note
that to apply this correction, we must first convert the quarterly compounded futures
rates into continuously compounded ones, using the formula f f u t (0, T, T + 0.25) =
4 × ln[1 + f4f u t (0, T, T + 0.25)/4]. Given the forward rates, we can compute the discount
using the same procedure as in Section 5.1.3 in Chapter 5. One intermediate step, however,
is necessary to compute discounts at a quarterly frequency. In particular, we interpolate
the semi-annual forward rates from Table 6.13 to obtain quarterly forwards, reported now
in the second column in Table 6.14. To avoid misunderstandings, it is important to recall
that the “Maturity” in Table 6.13 refers to the maturity of the futures contract. This implies
that a maturity T in Table 6.13 then corresponds to the forward rate from T to T + 0.25
in the last column. In contrast, in Table 6.14 the column “Maturity” refers to the time at
which the forward delivery actually takes place. The first entry, for instance, is simply
the continuously compounded LIBOR rate, which matures at T = 0.25. The entry corresponding to T = 0.5 is instead the interpolated value of the two first forward rates, 2.641%
and 2.690%, in the last column of Table 6.13, and so on. Given the forward rates, we can
finally compute the discounts up to T = 3 using Equation 5.15, as illustrated in Example
5.2 in Chapter 5.
For maturities longer than 3 years, we use swap rates to extract the LIBOR discount
curve. The original quotes are in Table 6.13. Column 3 of Table 6.14 reports linearly
interpolated swap rates for intermediate maturities at a semi-annual frequency, which is
the frequency of payments for the fixed rate payer in a plain vanilla swap. These swap
rates are converted into a discount factor Z(0, T ) starting with maturity T = 3.5 according
to Equation 5.45 in Chapter 5, for Δ = 0.5. The starting discount for the procedure is
Z(0, 3) = 90.223% obtained from the first part of the procedure. Since this procedure
only provides the discounts at semi-annual frequency, the remaining discounts at the other
quarterly dates are obtained by linear interpolation of the adjacent discounts. For instance,
Z(0, 4.25) = 0.5 × Z(0, 4) + 0.5 × Z(0, 4.5), where the latter two are instead computed
from the semi-annual swap rates. Column 4 in Table 6.14 reports the method or data used
to compute each of the discounts in column 5. Column 6 reports the implied spot rates.
The last three columns in Table 6.14 show the computation of the discount factors
Z(0, T ) and spot rates r(0, T ) for the case in which swap rates are used starting at T = 1
instead of T = 3.5, that is, for the shortest maturity swap data are available. As can
be seen, the spot rates obtained from swaps are always higher than those obtained from
12 The
adjustment in Equation 6.17 holds under a specific model, called the Ho-Lee model, which is developed in
Chapter 17.
13 The volatility σ can also be computed from options, as in Chapter 17.
236
INTEREST RATE DERIVATIVES: FUTURES AND OPTIONS
Figure 6.5 LIBOR Forward Curve when Using Futures and Swaps
5
Spot Curve
Forward Curve
4.5
Rate (%)
4
3.5
3
2.5
0
0.5
1
1.5
2
2.5
3
Time to Maturity
3.5
4
4.5
5
Eurodollar futures. The difference is attributed to a liquidity premium that characterizes
the over-the-counter swap contracts.
One issue with the methodology of computing the yield curve using securities with
different liquidity is that the forward curve may display an unusual behavior. For instance,
Figure 6.5 plots the spot curve (the solid line) and the forward curve (the dashed line)
obtained from the procedure in Table 6.14. Because of the sudden increase in the spot
rate at maturity T = 3.5, which is due to the change from the use of futures to swaps, the
forward curve displays a sudden bump, which is clearly an artifact of the change in liquidity
of the instrument. A methodology to resolve the problem is to smooth out the bump in the
forward curve, and obtain the yield out of the smoothed forward curve.
Table 6.14
Maturity
2.760
2.668
2.813
3.009
3.192
3.366
3.550
3.712
3.844
3.970
4.090
4.180
4.247
4.315
4.388
4.447
4.491
4.539
4.595
Interpolated
Swap Rates
2.861
3.027
3.193
3.340
3.487
3.603
3.720
3.807
3.894
Method
Discount
Spot Rate
Method
Swap-Based
Discount
Swap-Based
Spot Rate
LIBOR
FUT
FUT
FUT
FUT
FUT
FUT
FUT
FUT
FUT
FUT
FUT
FUT
SWAP
INTERP
SWAP
INTERP
SWAP
INTERP
SWAP
99.312
98.652
97.961
97.227
96.454
95.646
94.800
93.925
93.027
92.108
91.171
90.223
89.270
88.182
87.190
86.198
85.234
84.271
83.290
82.309
2.760
2.714
2.747
2.813
2.888
2.968
3.051
3.134
3.213
3.288
3.361
3.430
3.492
3.593
3.656
3.713
3.759
3.803
3.849
3.894
LIBOR
FUT
FUT
SWAP
INTERP
SWAP
INTERP
SWAP
INTERP
SWAP
INTERP
SWAP
INTERP
SWAP
INTERP
SWAP
INTERP
SWAP
INTERP
SWAP
99.312
98.652
97.961
97.198
96.394
95.589
94.720
93.850
92.940
92.029
91.068
90.107
89.148
88.188
87.196
86.204
85.241
84.277
83.296
82.316
2.760
2.714
2.747
2.842
2.938
3.007
3.100
3.174
3.254
3.322
3.402
3.472
3.535
3.591
3.654
3.711
3.757
3.801
3.848
3.892
APPENDIX: LIQUIDITY AND THE LIBOR CURVE
0.250
0.500
0.750
1.000
1.250
1.500
1.750
2.000
2.250
2.500
2.750
3.000
3.250
3.500
3.750
4.000
4.250
4.500
4.750
5.000
Interpolated
Forward
LIBOR CURVE on May 5, 2008
237
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CHAPTER 7
INFLATION, MONETARY POLICY, AND THE
FEDERAL FUNDS RATE
A treatment of interest rate determination and bond pricing has to take into account the
impact of monetary policy on interest rates. In this chapter we briefly review the Federal
Reserve System (the Fed) and the tools of monetary policy. In particular, we discuss the
way the Federal Reserve sets its main instrument of monetary policy, namely, the Federal
funds rate. The importance of the Federal funds rate, which is an overnight rate, in affecting
other short-term interest rates is apparent in Panel A of Figure 7.1, which plots the Fed
funds rate along with other short term interest rates, such as the three month T-bill rate and
the three month Eurodollar rate.1
7.1
THE FEDERAL RESERVE
The Federal Reserve is in charge of conducting U.S. monetary policy. The goals of the
monetary policy are spelled out in the Federal Reserve Act (Section 2a), which states that
the Federal Reserve should seek “to promote effectively the goals of maximum employment,
stable prices, and moderate long-term interest rates.” The Federal Reserve has only limited
power in affecting prices, employment, and interest rates, and the mechanism by which
it can do so is far from perfect. Indeed, although nowadays most economists agree that
the Federal Reserve is able to affect prices and real economic activity, there is still a large
1 Clearly,
the plot per se does not entail a causal relation between Fed funds rate and the other short-term rates, but
there is a large agreement that short-term rates react to monetary policy. See e.g., Cochrane and Piazzesi (2002).
239
240
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
Figure 7.1 Federal Funds Rate and other Interest Rates
Panel A. Federal Funds Rate and Short−Term Rates
20
Federal Funds
3−month T−bill
3−month Eurodollar
Rate (%)
15
10
5
0
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
Panel B. Federal Funds Rate and Long−Term Treasury Rates
20
Federal Funds Rate
3−year Treasury Spot Rate
5−year Treasury Spot Rate
Rate (%)
15
10
5
0
1965
1970
Source: Federal Reserve.
1975
1980
1985
1990
1995
2000
2005
2010
THE FEDERAL RESERVE
241
disagreement on the exact economic mechanism at work, as well as the size of the impact
of the Federal Reserve’s intervention in the market. One simple way to think about the
issue is to consider the Federal Reserve as trying to pilot a little sailboat in the middle of
major storm: By skillfully steering the boat’s wheel right and left, it is possible to give
some directions to the sailboat and perhaps avoid large underwater rocks. However, waves,
winds and currents of various sorts may eventually determine the ultimate course of the
boat. Indeed, as some sailors know, sometimes “too much” steering and wrong moves may
even cause the sailboat topple over and sink, whereas less proactive piloting might save
the boat, if not its direction. Piloting the economy safely through the waves of economic
booms and recessions is most definitely a difficult, and sometimes impossible, task for the
Federal Reserve, especially given the limited tool set at its disposal. We now turn to discuss
the tools of monetary policy, and their impact on interest rates.
7.1.1
Monetary Policy, Economic Growth, and Inflation
The Federal Reserve controls the money supply, that is, the total amount of money available
in the economy. The total money supply not only depends on the total amount of physical
currency available in the economy – i.e., the total amount of dollar bills and coins – but
also on the additional money that is created through the banking system. To grasp the idea,
suppose you win $1 million at the lottery, and the Federal Reserve prints out this amount
of money to pay you the prize. Most likely, you will not keep the $1 million at home,
but you will deposit it in a bank. The bank itself may keep about 10% of it in reserves at
the Federal Reserve bank, and lend the remaining $900,000 to a homeowner, for instance,
who may use it to purchase a home. The home seller receives the $900,000 and likely
will deposit this amount in another bank, which will retain another 10% of this deposit as
reserves, and lend the rest to other borrowers. And so on. That is, the initial $1 million of
physical currency printed out by the Federal Reserve generates far more currency that can
be used for transactions. The total money supply is then a multiple of the available physical
currency, a multiple that is generated by the banking system through the lending/borrowing
channel. How large is this multiple? If each bank keeps the same fraction of deposits as
reserves, e.g., 10% in the previous example, and if all of the loans end up going into other
deposits, then the answer is actually surprisingly simple: It is simply the ratio of deposits
over reserves, that is, 10 in the previous example.2 In other words, if banks keep 10%
of deposits in reserves, the total money supply is ten times the total amount of currency
printed by the Federal Reserve. This multiple is called the money multiplier.
It follows from the above discussion that the Federal Reserve affects the total money
supply through both the amount of currency that it actually prints and the reserve requirements, which in turn affects the money multiplier. However, note that if you did not deposit
the $1 million in a bank to start with, but decided to keep it at home in your own safe, then
the channel is broken. Similarly, if the bank decided to keep a larger fraction of deposits in
reserves, the multiplier is smaller. Therefore, the relation between the amount of currency
printed by the Federal Reserve and the actual money supply depends crucially on the desire
of people to keep currency in their pockets compared to depositing it in a bank, and the
amount of reserves that banks decide to keep in addition to any regulatory requirement.
2 To
see this, let f be the fraction of deposits kept in reserves. Then, the total amount of money created by an
∞
additional dollar is $1 + $1 × (1 − f ) + [$1 × (1 − f )] × (1 − f ) + ... = $1 ×
(1 − f )j = $f1 .
j=0
242
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
That is, the Federal Reserve does not have full control of the money supply, but it can only
affect it through its monetary policies decisions, as further discussed in the next section.
How does money supply affect real output? In the short run, an unexpected increase in
money supply tends to lower interest rates, as there is a temporary excess of funds available
for lending. A lower interest rate in turn stimulates aggregate demand, as individuals may
decide to borrow more money to spend it for consumption (e.g., purchase a new car or a
new home), while firms find it cheaper to borrow money for investment purposes, thereby
increasing the demand for intermediate goods or machinery. This stimulus to the aggregate
demand increases real output, and decreases unemployment, as firms need to hire new
workers to produce more to meet the increased aggregate demand. Vice versa, a tightening
of the money supply increases interest rates, which induce individuals to save more and thus
consume less, and firms to decrease investments as it costs more to borrow. As aggregate
demand declines, so does real output.
Why doesn’t the Federal Reserve then keep expanding the money supply to foster
growth and employment? Unfortunately, while an increase in money supply stimulates the
economy in the short run, it also increases inflation, which instead has a detrimental effect
on economic growth in the long run. Indeed, while at the short horizon there is a tradeoff
between low unemployment and high inflation,3 it turns out that at the long horizon an
increase in money supply growth only generates higher inflation, with not much gain in
real economic growth. Thus, the Federal Reserve tends to conduct its monetary policy on
a temporary basis, increasing money supply during recessionary periods to stimulate the
economy, but tightening the money supply when the economy expands too quickly to keep
inflation under control. Indeed, a long expansionary monetary policy tends to increase
investors’ expectation of future inflation, which in turn increases long-term nominal rates
and decreases economic activity. Section 7.2 below further discusses the relation between
inflation, employment growth, and monetary policy, while Section 7.4 investigates the
relation between inflation and nominal rates. We now turn in the meantime to describe the
tools of monetary policy.
7.1.2 The Tools of Monetary Policy
The Federal Reserve has three main tools of monetary policy:
1. Open market operations, which are interventions in the market to buy or sell
Treasury securities.
2. Reserve requirements, which are the amount of reserves that depository institutions
(banks) are required to have at the Federal Reserve Bank.
3. The Federal discount rate, which is the rate at which the Federal Reserve lends to
FDIC-approved depository institutions.
By far the main tool of active monetary policy is the first one, open market operations,
which are decided by the Federal Open Market Committee (FOMC). The reserve requirement is the responsability of the Board of Governors of the Federal Reserve, and the
3 The short-run negative relation between unemployment and inflation is typically referred to as the Phillips curve,
after the economist A. W. Phillips who first showed this relation in British data in 1958.
THE FEDERAL RESERVE
243
Federal discount rate is set by the boards of directors of the Federal Reserve banks, subject
to review and determination by the Board of Governors.
7.1.3
The Federal Funds Rate
The Federal Open Market Committee meets about eight times a year, according to a
predetermined calendar, to discuss its monetary policy actions. The main instrument the
Federal Reserve uses to have an impact on the economy is altering the Federal funds
rate. The Federal funds rate is the rate at which depository institutions can borrow or lend
overnight reserves at the Federal Reserve Bank. Each depository institution is required to
keep reserves with the Federal Reserve – about 10% of the amount of its deposits. Because
of its daily operations a depository institution may run a deficit of reserves, and rather than
simply replenish them, it may be cheaper to borrow such funds from another depository
institution that may be running a surplus. The Federal funds rate is the rate at which these
borrowing and lending transactions occur.4 The incentive to keep reserves at the Federal
Reserve Bank close to the limit stems from the fact that these reserves pay an interest that
is lower than the rate of available alternative investments, and therefore it is costly to hold
large reserves. Indeed, up until October 2008 such reserves actually paid zero interest.
The Federal funds rate itself then is not decided by the Federal Reserve, but it is an
equilibrium level resulting from lending and borrowing of reserves at the Federal Reserve
Bank. However, open market operations have a large impact on this equilibrium rate, as
they affect the total amount of reserves available to depository institutions. For instance,
if the Fed buys Treasury securities in an open market operation from one of the depository
institutions, it pays for them by crediting the depository institution’s account at the Federal
Reserve. Because the depository institution can then lend these cash balances to other banks
at the Fed funds rate, the open market operation effectively increases the total supply of
cash balances available for lending, which in turn tends to decrease the equilibrium market
clearing Fed funds rate. Vice versa, if the Fed sells Treasury securities to a depository
institution, it effectively drains the total available cash balances of the depository institutions
available for lending at the Fed funds rate. In this case, the equilibrium Fed funds rate
tends to increase.
At FOMC meetings, the Federal Reserve decides a target Federal funds rate, and
then changes the size of open market operations or their direction to keep the effective
equilibrium Federal funds rate close to the target Federal funds rate. If the Federal Reserve
wants to have a permanent impact on the Fed funds rate, then it carries out open market
operations by an outright purchase or sale of Treasury securities, which will permanently
affect the total amount of reserve balances available to depository institutions. However,
more often the Federal Reserve only wishes to counterbalance temporary or seasonal
variations in the total supply of reserves. In this case, the Federal Reserve carries out
open market operations through repurchase and reverse repurchase agreements, that is,
contractual agreements in which the Federal Reserve sells (or buys) Treasury securities to
(or from) primary dealers with the agreement to buy them back (or sell them back) at an
4 The
Federal funds rate is not the rate at which banks can borrow from the Federal Reserve. That one is called
the discount rate. Also, depository institutions differ in terms of credit rating, and so the rate at which these
transactions occur differs across institutions. The reported rate is the one pertaining to the depository institutions
with the highest creditworthiness.
244
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
agreed upon future date, such as the following day (overnight repo) or a couple of weeks
(long-term repo). For more details about repo transactions, see the discusion in Chapter 1.
Section 7.7 below further discusses the workings of U.S. monetary policy in the context of
the subprime credit crisis of 2007 - 2008. This financial crisis, the largest in U.S. history
since the Great Depression, saw the Federal Reserve at the center stage, as it used all of the
monetary policy tools at its disposal – as well as new ones – in its attempt to prevent the
U.S. economy from entering into a long and deep recession.
It should be mentioned that because the Federal funds rate is a borrowing rate among
depository institutions, it is higher than the short-term Treasury bill rate, as it incorporates
a little premium for default. For the same reason, it is also higher than the repo rate, the
rate for collateralized borrowing. Indeed, the Fed funds rate is closer to the LIBOR, which
as we know from Chapter 1 is the interbank uncollataralized borrowing rate.
7.2 PREDICTING THE FUTURE FED FUNDS RATE
The prediction of future interest rate movements is a hard task. Indeed, in the short run,
say daily or weekly, the change in interest rate is almost unpredictable, and the best guess
of the interest rate tomorrow is the interest rate today. For the medium-to-long horizon, in
contrast, interest rate movements up and down are predictable to some extent. For instance,
from Figure 7.1 we see that in 2003 the Federal funds rate hit its lowest level in past history,
at less than 1%.5 Simple reasoning that such a scenario was very unusual for the Federal
funds rate led many market participants to believe that the interest rate would soon increase
again (as in fact it did). Many observers, however, were worried that the Federal Reserve
could get stuck in a Japan-style zero interest rates’ environment. That is, the fact that the
Fed funds rate had been higher in the past was no guarantee that it would increase again in
the future.
Let’s look at the relation between the Federal funds rate and some important macroeconomic variables, such as inflation and employment.
7.2.1 Fed Funds Rate, Inflation and Employment Growth
According to its mandate, the Federal Reserve must promote employment, stable inflation
and low long-term interest rates. It is then useful to see whether there is any relation
between these variables and the main tool of Federal Reserve intervention, the Federal
funds rate. Starting with medium-to-long term Treasury rates, Panel B of Figure 7.1 plots
the Fed funds rate and the 3-year and 5-year zero coupon Treasury spot rates. Indeed,
the variation of these medium-to-long term yields is correlated with the variation of the
overnight Federal funds rate, although this correlation is much less pronounced than the
one of short-term rates, as depicted in Panel A.
Figure 7.2 shows the relation between the Federal funds rate, the annual inflation level
(Panel A) and employment growth (Panel B), as proxied by Nonfarm payroll growth rate.
We use this variable over the many other employment related variables, as it appears the
most correlated with the variation of the Fed funds rate. Panel A shows that indeed as
inflation increased and decreased in the 1960s and 1970s, the Federal funds rate followed
5 In
December 2008, however, the Federal funds rate hit essentially zero, beating the record low level of 2003.
245
PREDICTING THE FUTURE FED FUNDS RATE
Figure 7.2 Federal Funds Rate, Inflation and Employment
Panel A. Federal Funds versus Annual CPI Inflation
Fed Funds / Inflation (%)
20
Federal Funds
CPI Annual Inflation
15
10
5
0
1965
1970
1975
1980
1985
1990
1995
2000
2005
2010
Panel B. Federal Funds versus Annual Growth in Nonfarm Payroll
20
Fed Funds (%)
16
12
8
12
8
4
4
0
0
1965
1970
1975
1980
1985
1990
1995
2000
2005
Payroll Growth Rate (%)
16
Federal Funds
Payroll Annual Growth
−4
2010
Source: Federal Reserve and Bureau of Labor Statistics.
it quite closely. In the 1980s, in particular, the Federal funds rate remained high for a
substantial amount of time while inflation was not under control. The connection between
inflation and the Federal funds rate is attenuated in the 1990s, as the low inflation appears
to be considered under control. Indeed, Panel B shows that in the 1990s and 2000s, the
Fed funds rate is moving in sync with the annual growth rate in Nonfarm payroll.
Of course this analysis is only suggestive as the conduct of monetary policy is very
complex: In particular, every month the Federal Reserve observes hundreds of indicators
about the economy and it must filter the information that these indicators provide so as to
decide the best course of action in terms of monetary policy and interest rate determination.
The sheer amount of data is daunting: they include economic growth indicators, price
indices of various sorts, employment indicators, a whole host of financial variables, and the
health of the banking system. In addition, the Fed avails itself of sophisticated econometric
techniques that have been put forward by economists and statistician to squeeze the most
important information out of these many variables.
246
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
We can illustrate the relation between the Fed funds rate, inflation and employment
by looking at the variables in Figure 7.2. Let us start from the easiest possible way of
predicting future Fed funds rates. As it is clear from Figure 7.2, the Fed funds rate, as does
any other rate, goes up and down, and it hovers around 6.57%, which is its average during
the period. Thus, if we see that the Fed funds rate extremely low, such as 1% as in 2003,
we can reasonably expect that it will go back up, as soon as economic conditions improve,
or inflation starts moving up again. Similarly, if the interest rate is very high, we could
reasonably expect it will sooner or later decline again, as soon as inflation is under control.
Therefore, the simplest way to predict the Fed funds rate is to try to forecast it using its
current level. Namely, we can run the following regression:
rF F (t + 1) = α + β 1 × rF F (t) + (t + 1)
(7.1)
The coefficient β 1 tells us the relation between the Fed funds rate at time t + 1 given
its value at t. Using the data in Figure 7.2 Panel A of Table 7.1 shows the regression
results, where the horizon column is the predicting horizon. For instance, the first row
shows the predictability of the Fed funds rate one month ahead. The second row shows the
predictability of the Fed funds rate three months ahead. And so on. The columns headed
“α” and “β” show the numerical values estimated from the regression in Equation 7.1,
while the following two columns report the estimates’ standard errors (se).6 Finally, the
last column reports the regression R2 , which is a number between 0 and 1 describing how
good the right-hand side variable in Equation 7.1 is in predicting the left-hand side variable.
For instance, the table shows that it is much easier to accurately predict the interest rate in
one month (R2 = 96.66%) than in one year (R2 = 57.59%).
As discussed, the Fed funds rate reacts also to the labor market conditions and the
inflation rate. It appears sensible then to try to forecast the future Fed funds rate by using
information also from the business cycle. In particular, Panel B of Table 7.1 reports the
results of the following regression:
rF F (t + 1) = α + β 2 × X P ay (t) + β 3 × X I n f (t) + (t + 1)
(7.2)
where X P ay (t) is the annual growth in nonfarm payrolls, and X I n f (t) is the annual
growth in the CPI index. Both these series are plotted in Figure 7.2. The regression makes
it apparent that the Federal Reserve reacts to labor market conditions and inflation. Both
coefficients β 2 and β 3 are positive, showing that a decrease in the growth of rate in nonfarm
payroll tends to generate a decrease in the Fed funds rate, while an increase in the inflation
tend to increase the Fed funds rate, as intuition would have it.
Inflation and nonfarm payroll growth explain a good deal of the variation in the Fed
funds rate, about 50% or so, but not all of it. In particular, the R2 is lower than in Panel
A, in which the only regressor was the Fed funds rate itself. There are two reasons for
this: First, the Federal Reserve does not only react to payroll growth and inflation, but
it considers a whole host of macroeconomic variables, as discussed earlier. Second, the
regression analysis above has at its core assumption that the parameters β 2 and β 3 remain
constant for the whole sample used in the analysis, in this case the 40 year sample period
from 1968 - 2008. The Federal Reserve monetary policy has likely changed over the years,
6 In
this and the following tables, the standard errors are computed using the Newey-West adjustment for autocorrelation and heteroskedasticity in the errors. For robustness, only nonoverlapping data are used in the estimation.
For instance, for the annual forecasting regression we only use data at the annual frequency.
PREDICTING THE FUTURE FED FUNDS RATE
247
and therefore forcing the parameters to remain constant decreases the ability of the model
to predict future interest rates. Sophisticated econometric techniques that take into account
the variations in these parameters have been develped, but we will not explore them here,
as they are beyond the scope of this chapter.
Instead, we can check whether the past Fed funds rate rF F (t) is a sufficient predicting
variable of future Fed funds rates, or whether adding information from the macroeconomy
helps. We therefore run the regression including both the past Fed funds rate and the
macroeconomic variables discussed earlier:
rF F (t + 1) = α + β 1 × rF F (t) + β 2 × X P ay (t) + β 3 × X I n f (t) + (t + 1)
(7.3)
Panel C of Table 7.1 reports the results. The results indeed show that adding macroeconomic information helps increase the predictive power of the regression. The last
column in Panel C reports the “adjusted R2 ,” which like the R2 measures the ability of
the right-hand side variables in Equation 7.3 to predict the left-hand side variable, but it
also corrects for the number of regressors used. For instance, if we add a regressor on the
right-hand side variable that is not helping at all to forecast the Fed funds rate, the adjusted
R2 declines. As can be seen, this last number is substantially higher than the R2 ’s in both
Panel A and B.
Which variable is more important to predict future Fed funds rate, payroll growth or
inflation? Once we account for lagged Fed Funds rate in the regression, we find that the
coefficient on the inflation variable β 3 is not significant. This means that its standard errors
[Column “se(β 3 )”] are so large that the coefficient β 3 cannot statistically be considered any
different from zero. That is, the fact that it is different from zero may be due just to chance,
to the particular sample used. This does not mean that the Federal Reserve does not react
to inflation. Indeed, recall that β 3 was instead different from zero in Equation 7.3. The
lack of significance implies that lagged Fed funds rate contains all of the information about
future Fed funds rates that is also included in the inflation rate itself. That is, the current
inflation rate does not add anything to the prediction of future Fed funds rates above and
beyond the current level of Fed funds itself. Instead, payroll growth does add information
to forecast future Fed funds rates.
7.2.2
Long-Term Fed Funds Rate Forecasts
How can we use the information in Table 7.1 to forecast future interest rates? Suppose it is
F
= 2.98%. According to
February 2008. The current level of the Fed funds rate is rFF eb08
the model in Equation 7.1, the forecasted rate one month ahead would be
FF
FF
rM
ar 08 = α + β × rF eb08 = 0.1019 + 0.9840 × 2.98% = 3.0342%
where we set (t + 1) = 0 in our forecast, because its expected value is in fact zero. To
compute the two-months-ahead value, we need to insert the predicted one-month-ahead
into Equation 7.1 again:
F
FF
rAF pr
08 = α + β × rM ar 08 = 0.1019 + 0.9840 × 3.0342% = 3.0876%
and so on. By repeating this procedure, we obtain the solid line in the top panel of Figure
7.3. This shows that the model predicts an increase in the future Fed funds rate. The reason
248
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
Table 7.1 Predicting the Fed Funds Rate: 1968 - 2008
Panel A: Fed Funds Rate on Past Fed Funds Rate
Horizon
α
β1
se(α )
se(β )
R 2 (%)
1m
3m
6m
1y
0.1019
0.5312
1.1433
1.5285
0.9840
0.9190
0.8255
0.7611
0.1191
0.2891
0.3946
0.7784
0.0220
0.0516
0.0676
0.1420
96.66
84.31
67.93
57.59
Panel B: Fed Funds Rate on Past Payroll Growth and Inflation
Horizon
α
β2
β3
se(α )
se(β 2 )
se(β 3 )
R 2 (%)
1m
3m
6m
1y
1.9946
1.7750
1.7134
1.1656
0.3386
0.4323
0.5499
0.8634
0.8675
0.8829
0.8468
0.8095
0.4711
0.6879
0.8213
0.9722
0.1558
0.2266
0.2777
0.2855
0.1020
0.1537
0.2064
0.2784
53.14
54.97
52.72
58.75
Panel C: Fed Funds Rate on Past Fed Funds Rate, Payroll Growth, and Inflation
2
Horizon
α
β1
β2
β3
se(α )
se(β 1 )
se(β 2 )
se(β 3 )
R 2 (%)
R (%)
1m
3m
6m
1y
-0.0196
0.1259
0.4162
0.2175
0.9594
0.8170
0.6411
0.5300
0.0731
0.2122
0.3528
0.7309
0.0326
0.1506
0.2833
0.319
0.097
0.1768
0.3313
0.7138
0.017
0.0576
0.0895
0.0988
0.0203
0.0521
0.1338
0.202
0.0271
0.0918
0.1879
0.2524
96.82
85.91
72.70
71.08
96.81
85.74
71.99
69.52
Notes: Coefficients in bold are statistically significant at 1% confidence level.
is that the current Fed funds rate is below its historical average, and therefore the model
forecasts that in the long run it will revert back to its long term average.
The other lines in Figure 7.3 report the outcome of the same forecasting exercise, but
we use the α and β obtained from the regression that uses quarterly data (the second row
in Panel A of Table 7.1), semi-annual data (the third row in Panel A) or annual data (the
fourth row in Panel A). These forecasts are similar to the ones obtained from the monthly
frequency: We should not expect to obtain exactly the same forecast, as different sampling
frequencies capture different dynamic aspects of the behavior of the interest rate.
7.2.2.1 Adding Macro Variables The long-term forecasts of the Fed funds rates
performed above only use information about the past Fed funds rate. As we have shown in
Panel C of Table 7.1 adding macroeconomic information helps to forecast future interest
rates. Therefore, it may be advisable to also use information from macroeconomic variables
to formulate long term forecasts. We can perform exactly the same exercise as in the
previous section, but now also include macroeconomic variables. In particular, given the
P ay
F
current value of Fed funds rate (rFF eb
08 = 2.98%), payroll growth (XF eb 08 = 0.6250%)
I nf
and inflation (XF eb 08 = 4.0380%), we can insert them into Equation 7.3 and obtain the
prediction of the Fed funds rate for March 2008 as
FF
rM
ar
08
F
= α + β 1 × rFF eb
08
ay
I nf
+ β 2 × XFP eb
08 + β 3 × XF eb 08
= −0.0196 + 0.9594 × 2.98 + 0.0731 × 0.6250 + 0.0326 × 4.0380
= 3.0167%
PREDICTING THE FUTURE FED FUNDS RATE
Figure 7.3
249
Long-Term Federal Funds Rate Forecasts
Panel A. Predicting the Fed Funds Rates with Past Fed Funds Rates
7
1 Month
3 Months
6 Months
12 Months
Interest Rate (%)
6
5
4
3
2
0
1
2
3
4
5
Years Ahead
6
7
8
9
10
Panel B. Predicting the Fed Funds Rates with Past Fed Funds Rates, Payroll Growth, and Inflation
8
1 Month
3 Months
6 Months
12 Months
Interest Rate (%)
7
6
5
4
3
2
0
1
2
3
4
5
Years Ahead
6
7
8
9
10
250
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
Unlike when we use use the past Fed funds rate to predict future Fed funds rates, we
now find a roadblock. In particular, we cannot proceed because in order to use the same
F
FF
equation to predict rAF pr
08 , we need to have a forecast not only of rM ar 08 , which we have,
P ay
I nf
but also of XM ar 08 and XM ar 08 , which we do not have.
How do we move forward then? The solution is to use the same three variables rF F (t),
P ay
(t), and X I n f (t) to also forecast future payroll growth and future inflation. In other
X
words, we can run the following two additional regressions:
X P ay (t + 1) = αP ay + β P1 ay rF F (t) + β P2 ay X P ay (t) + β P3 ay X I n f (t) + P ay (t + 1)
X I n f (t + 1) = αI n f + β I1 n f rF F (t) + β I2 n f X P ay (t) + β I3 n f X I n f (t) + I n f (t + 1)
Given the parameter estimates (not reported), we can then proceed exactly as in the previous
section: Given the three values of the right-hand side variables at any time t, rF F (t),
X P ay (t), and X I n f (t), we can use the regressions to compute their values at t + 1. Given
the value at t + 1, we can compute the value at t + 2, and so on.
Panel B of Figure 7.3 reports the results for the Fed Funds rate forecast. Consider the
solid line, which uses estimates obtained from monthly frequency. The forecast is similar
to the one obtained in Panel A using only the Fed funds rate, but it now presents a difference
in terms of dynamics. In particular, the Fed Funds rate is predicted to increase slowly at
first, to pick up its rate of increase up to the maximum of about 7%, and then decline back
to its long-term average of 6.5%. Eye balling the historical variation in the Fed funds rate
in Figure 7.2, and especially the last few decades, we see that indeed its dynamics over
time are characterized by periods in which it is stable at some value, and then suddenly
increases or decreases for a few consecutive months. This pattern is partly captured by the
joint dynamics of Fed funds rate, payroll growth, and inflation.
7.2.3 Fed Funds Rate Predictions Using Fed Funds Futures
Fed funds futures have been trading on the CBOT since 1988.7 The description of the
contract is in Table 7.2. As discussed in Chapter 6, if a trader enters into the Fed funds
futures at a futures rate f F u t (t; T ), the payoff is approximately8
Payoff at T = $5 million × rF F (T ) − f F u t (t; T )
(7.4)
Intuitively, if some traders think that the Federal Reserve will increase the Fed funds
target rate at the next meeting (or earlier), then they would expect a positive payoff from
Equation 7.4. If many traders have these beliefs, then they would bid up the Fed funds
futures f F u t (t; T ). We should then expect that a high Fed funds futures would be correlated
with a high future Fed funds rate. Panel A in Table 7.3 contains the results of the regression:
rF F (t + h) = α + β × f F u t (t, t + h) + (t + h)
(7.5)
where h is the Fed funds futures horizon at initiation.9
7 The
Chicago Board of Trade (CBOT) is now part of the Chicago Mercantile Exchange (CME) group.
profits and losses accrue during the life of the futures, and therefore we must take into
account the time value of money. Such calculations are neglected in Equation 7.4.
9 These results are obtained from using the “Generic” Federal funds futures data from Bloomberg, Inc, for various
maturities h = 1, 2, .., 6 . The “Generic” futures contract with maturity h is the contract that is closest to the
8 As in any futures contract,
PREDICTING THE FUTURE FED FUNDS RATE
Table 7.2
251
30-Day Fed Funds Futures (Chicago Board of Trade)
Contract Size
$5 million
Tick Size
$20.835 per 1/2 of one basis point (1/2 of 1/100 of one percent of $5 million on a 30-day basis
rounded up to the nearest cent).
Price Quote
100 minus the average daily Fed funds overnight rate for the delivery month (e.g. a 7.25 percent
rate equals 92.75).
Contract Months
First 24 calendar months
Last Trading Day
Last business day of the delivery month. Trading in expiring contracts closes at 2:00 pm, Chicago
time on the last trading day.
Settlement
The contract is cash settled against the average daily Fed funds overnight rate, rounded to the nearest
one-tenth of one basis point, for the delivery month. The daily Fed funds overnight rate is calculated
and reported by the Federal Reserve Bank of New York.
Trading Hours
Open Auction: 7:20 am - 2:00 pm, Central Time, Monday - Friday
Electronic: 5:30 pm - 4:00 pm, Central Time, Sunday - Friday
Ticker Symbols
Open Auction: FF
Electronic: ZQ
Daily Price Limit
N/A
Source: CBOT Web site, http://www.cbot.com/cbot/pub/cont detail/0,3206,1525+14446,00.html,
accessed on September 10, 2008.
252
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
The results in Panel A of Table 7.3 show that indeed there is a good deal of predictability
of the future Fed funds rates from Fed funds futures. The R2 are relatively high across
maturities, even for the one-year ahead forecast horizon. Since the sample is different, we
cannot compare the results in this table with those in Table 7.1. It is illustrative to report
the same results discussed earlier, but for the shorter sample 1989 - 2008. Panel B reports
the result of the regression in Equation 7.1. Comparing with the longer period (Panel A
in Table 7.1) we see that during the past two decades the Fed funds rate has become more
persistent (i.e. less variable), and thus easier to predict. The R2 s are uniformly higher in
the shorter sample than in the longer sample. Comparing now Panels A and B of Table
7.3 we see that the Federal funds futures indeed improves upon the past Fed funds rate in
predicting the future Fed funds rate, especially for the longer forecasting horizon. Market
expectations about monetary policy are reflected in current prices – especially the Fed funds
futures – which in turn help predict the future movement in interest rates.
Interestingly, Panel C of Table 7.3 shows that the level of predictability obtained using
the Fed funds futures alone (Panel A) is similar to the one we obtained using the past
Fed funds rate, payroll growth and inflation. Indeed, to see whether the Fed funds futures
provides any additional information compared to the other three variables, we can include
it in the predicting regression as well. Panel D of Table 7.3 performs this exercise, and
shows that indeed nonfarm payroll growth is still a significant predictor of Fed funds rates,
even after considering the information from the Fed funds futures itself. Although a topic
of subsequent chapters, one way to interpret this result is to realize that it is not exactly
accurate to consider the Fed funds futures as an unbiased forecast of future interest rates.
Indeed, the Fed funds futures contains a mix of market forecast of the Fed funds rate as
well as market participants’ risk attitude towards speculation in the futures market. This
risk attitude is also reflected in the futures, thereby biasing the forecast (see Example 7.2
below for a similar argument in the case of forward rates). If this bias depends on market
conditions, such as the business cycle, then we would conclude that adding back a business
cycle related variable, such as payroll growth, may add forecasting power to the Fed funds
futures itself.10
In Table 7.3 we have four ways of forecasting future Fed Funds rates: (a) using only
the past Fed funds rate; (b) adding to it macroeconomic information; (c) using Fed funds
futures; and (d) all of the above. Are these forecasts any different, or do they look alike,
as we saw in Figure 7.3? Figure 7.4 plots the forecasts of future Fed funds rates as of
February, 2008 for these four cases up to a one-year horizon. As can be seen, when we use
only past information, the forecasting model tends to generate an upward forecast [case (a)
and (b)]. In contrast, when we also consider the information from the Fed funds futures
[case (c) and (d)], the market is forecasting an additional decrease in the Fed funds rate
(as it indeed happened). In retrospect, part of the reason for the different forecasts stems
from the particular circumstances surrounding the economy at the beginning of 2008. As
discussed in Section 7.7 below, the U.S. economy was undergoing a major credit crisis,
and the Federal Reserve was worried about the impact that failing banks might have on the
desired maturity. The futures price is rolled over from the previous month. Because the regression uses monthly
data, the fact that Fed funds futures has monthly maturities does not generate any bias in the estimates. The only
caveat is for the annual regression h = 12: Because the Fed funds futures appears to lack liquidity for such a
long horizon, we used the 6-month futures h = 6 in the regression instead.
10 See Piazzesi and Swanson (2008) for a related discussion.
Table 7.3 Predicting the Fed Funds Rate: 1989 - 2008
Panel A: Fed Funds Rate on Fed Funds Futures
Horizon (Months)
α
β
se(α)
se(β)
R2
1
3
6
12
-0.0032
-0.0510
-0.0101
0.9549
0.9943
0.9895
0.9530
0.7214
0.0148
0.0794
0.2291
0.5013
0.0038
0.0171
0.0384
0.0723
99.72
98.20
91.12
65.79
Panel B: Fed Funds Rate on Past Fed Funds Rate
Horizon (Months)
α
β
se(α)
se(β)
R2
1
3
6
12
0.0352
0.1876
0.5181
1.4685
0.9866
0.9450
0.8597
0.6245
0.0443
0.1607
0.3435
0.5069
0.0086
0.0263
0.0504
0.08
99.09
94.87
83.68
51.19
2
Horizon (Months)
α
β1
β2
β3
se(α)
se(β 1 )
se(β 2 )
se(β 3 )
R
1
3
6
12
0.0024
0.0464
0.2203
1.2677
0.9546
0.8437
0.6431
0.2895
0.0958
0.2840
0.5511
0.8719
0.0159
0.0708
0.1752
0.1603
0.0541
0.1254
0.2367
0.3900
0.01
0.0278
0.0483
0.1183
0.0173
0.0454
0.0672
0.1117
0.0224
0.0553
0.1084
0.1635
99.35
97.10
92.35
73.34
Panel D: Fed Funds Rate on Past Fed Funds Rate, Payroll Growth, Inflation, and Futures
2
α
β1
β2
β3
β4
se(α)
se(β 1 )
se(β 2 )
se(β 3 )
se(β 4 )
R
1
3
6
12
-0.0112
-0.0525
-0.0295
0.9981
0.1395
0.0948
0.0419
-0.7180
0.0242
0.1246
0.3885
0.5711
0.0062
0.0212
0.1113
0.0925
0.8454
0.8445
0.7298
1.1813
0.0289
0.1074
0.2259
0.3871
0.0586
0.1350
0.2941
0.7106
0.0089
0.0386
0.0814
0.1211
0.0112
0.0376
0.0951
0.1122
0.0603
0.1426
0.3405
0.7864
99.73
98.46
94.33
79.65
Notes: Coefficients in bold are statistically significant at 1% confidence level.
253
Horizon (Months)
PREDICTING THE FUTURE FED FUNDS RATE
Panel C: Fed Funds Rate on Past Fed Funds Rate, Payroll Growth, and Inflation
254
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
Figure 7.4 Federal Funds Rate Forecasts
3.6
Past Fed Funds Rate
Past Fed Funds Rate, Payroll, Inflation
Fed Funds Futures
All
3.4
3.2
Fed Funds Rate (%)
3
2.8
2.6
2.4
2.2
2
1.8
1.6
0
0.1
0.2
0.3
0.4
0.5
0.6
Forecasting Horizon
0.7
0.8
0.9
1
long-term growth of the economy. Credit conditions are not part of the macro-economic
variables used in the regression analysis, while such “soft” information is embedded in the
Fed funds futures.
7.3 UNDERSTANDING THE TERM STRUCTURE OF INTEREST RATES
Why does the term structure of interest rates tend to slope upwards? What is the risk
involved in investing in Treasury securities? Can we predict medium-to-long term yields?
What about returns? In this section we look at some generic relations that have to hold
across yields, and the empirical evidence about their behavior over time. We begin with
one example.
EXAMPLE 7.1
Suppose today the continuously compounded 1-year spot rate is 3%. Assume that
we have perfect foresight and we know that next year the 1-year spot rate will be 5%.
What then should today’s 2-year yield be? To answer this question, we begin in the
future and work backward. If we know for sure that next year the 1-year yield will
be 5%, we also know that the price of a zero coupon bond next year will be
Pz (1, 2) = e−r (1,2)×1 × 100 = e−5% × 100 = 95.1229
Discounting this price to today, we have that the price today of a 2-year zero coupon
bond is
Pz (0, 2)
= e−r (0,1)×1 × Pz (1, 2) = 0.970445 × 95.1229 = 92.3116
Today’s 2-year yield is then r(0, 2) = − ln(.923116)/2 = 4%, the average between
today 1-year rate, 3% and next year 1-year rate, 5%. Is this surprising?
UNDERSTANDING THE TERM STRUCTURE OF INTEREST RATES
255
Note that we can rewrite
Pz (0, 2)
= e−r (0,1)×1 × Pz (1, 2) = e−r (0,1)×1 × e−r (1,2)×1 × 100
= e−r (0,1)−r (1,2) × 100
Because from the definition of the 2-year yield
Pz (0, 2) = e−r (0,2)×2 × 100
equating the right-hand sides of the last two equations implies that under perfect
foresight
r (0, 2) × 2 = r (0, 1) + r (1, 2)
or
1
1
r (0, 1) + r (1, 2)
(7.6)
2
2
The long-term yield is a weighted average of the current short-term yield and the
short-term yield next period.
r (0, 2) =
This example shows that if market participants are perfectly certain about the next year’s
1-year rate, then the 2-year yield is a weighted average of today’s and next year’s 1-year
rates. In other words, if market participants are certain that next year rates will be higher
than today’s, then the today’s yield curve will reflect this information by sloping upward.
Similarly, if market participants are certain that next year rates will be lower than today’s,
then today’s yield curve slopes downward.
This positive relation between market participants’ expectations about future rates and
the current shape of the yield curve goes under the name of expectation hypothesis. Note,
though, that market participants’ expectations of future rates are not the only determinants
of the current shape of the term structure of interest rates. We now introduce a simple
model illustrating other factors that affect the term structure of interest rates.
7.3.1
Why Does the Term Structure Slope up in Average?
In this section we highlight the importance of investors’ risk aversion in determining the
shape of the term structure of interest rates. The intuition is related to the risk of investing
in long-term bonds versus short-term bonds, a tradeoff already illustrated in Chapter 3
(see Section 3.3). In a nutshell, on average investors in the bond market are averse to
risk. As discussed in Chapter 3, longer-term bonds have a higher duration than short-term
bonds, and thus they are riskier. As a consequence, investors demand a higher yield to hold
long-term bonds over short-term bonds, thereby making the term structure of the interest
rate slope upward, on average. We now formalize this intuition within a formal model.
Let r(t, T ) be the continuously compounded yield between time t and time T . Let
today be t and as in Example 7.1 consider one-year-ahead predictions of future yields.
Let r(t + 1, T ) be the yield next year for the bond maturing at time T . This future yield
is obviously unknown to market participants at t. Assume that r(t + 1, T ) has a normal
distribution with mean Et (r(t + 1, T )) and variance Vt (r(t + 1, T )), where the subscript
t denotes that this expectation depends on the information up to t:
r(t + 1, T ) ∼ N (Et (r(t + 1, T )), Vt (r(t + 1, T )))
(7.7)
256
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
For given yield r(t + 1, T ), the value of a zero coupon bond at time t + 1 with maturity T
will be
(7.8)
Pz (t + 1, T ) = e−r (t+1,T )×(τ −1) × 100
where τ = T − t is time to maturity of the bond at t. What is the value today of the zero
coupon bond maturing at time T ? Since Pz (t + 1, T ) is not known today, we have
Pz (t, T ) = e−(r (t,t+ 1)+λ) × Et [Pz (t + 1, T )]
(7.9)
where λ denotes a risk premium for investing in long-term bonds for a one-year horizon
compared to safe 1-year zero coupon bonds. We discuss this premium further below. From
the properties of the log-normal distribution,11 we have
Pz (t, T ) = e−(r (t,t+ 1)+λ) × e−E t (r (t+ 1,T ))×(τ −1)+
( τ −1 ) 2
2
V t (r (t+1,T ))
× 100
(7.10)
Substituting also Pz (t, T ) = e−r (t,T )×τ × 100 we finally obtain the following decomposition for the long-term yield:
r (t, T )
1
(τ − 1)
× r (t, t + 1) +
× Et (r (t + 1, T ))
=
τ
τ
λ
+
τ
2
(τ − 1)
−
Vt (r (t + 1, T ))
2τ
(Expected future yield)
(Risk premium)
(Convexity)
(7.11)
Equation 7.11 shows the factors that affect the current long-term rate r(t, T ). The first
term in brackets on the right-hand side is the weighted average between the current shortterm rate, and the expected long-term yield next year. This is the same term appearing
on the right hand side of Equation 7.6 in Example 7.1, and simply says that if market
participants expect future long-term yields to be high, then the current yield is high as well.
The second term, λ, is a risk premium that market participants require to hold long-term
zero coupon bonds with maturity T over safe short-term bonds with maturity t + 1. To
understand the role of this term, note that we can rewrite Equation 7.9 equivalently as:
Pz (t + 1, T )
100
(7.12)
=
× eλ
Et
Pz (t, T )
Pz (t, t + 1)
The left-hand side is the expected gross return between t and t+1 from investing in the zero
coupon bond maturing at time T , while the term in square parenthesis on the right-hand
side is the return during the same period from investing in a zero coupon with maturity
t + 1. This latter return is known at time t and thus is riskless. Because eλ > 1 if and only
if λ > 0, Equation 7.12 says that the expected return during t and t + 1 on the long-term
bond is higher than the safe one-year return on a Treasury bill if and only if λ > 0. Higher
λ implies the long-term bond has a higher expected return compared to a riskless one-year
T-Bill return.
11 Given
#
$
x ∼ N (μ x , σ 2x ) and a constant A, we have E eA x = eA μ x +
A2
2
σ 2x
.
UNDERSTANDING THE TERM STRUCTURE OF INTEREST RATES
257
Why does a higher risk premium λ imply a higher yield to maturity? Because the
long-term bond pays no coupons, the only way a higher expected return can be achieved
between t and t + 1 is if the bond has a lower price at t. As we know, a lower price of the
bond today implies a higher yield, and the term λ in Equation 7.11 follows.
The last term in Equation 7.11 is related to the variance of the long-term yield r(t+1, T ),
and it is called convexity term. The source of this term is the nonlinear relation that exists
between the yield r(t + 1, T ) and the price Pz (t + 1, T ) = e−r (t+1,T )×(τ −1) × 100. Higher
volatility of the future yield implies a higher price, due to Jensen’s inequality, everything
else equal.12 Thus, a higher future yield volatility tends to decrease today’s yield. Although
it appears counterintuitive that higher volatility of future yields – which is related to risk
– decreases the current yield, we should recall the discussion in Section 4.1.4 in Chapter
4, according to which the same convexity implies that higher volatility of future yields
increases average returns, everything else equal. The lower yield counterbalances the
positive convexity effect on return.
The expression for the long-term yield in Equation 7.11 has some interesting implications, which are best described within the context of a simple example.
EXAMPLE 7.2
Consider again Example 7.1, but let r(0, 1) = 5%, and let market participants’
expectation of next year 1-year rate also be 5%: E[r(1, 2)] = 5%. Under perfect
foresight, r(0, 2) = 5%. However, if r(1, 2) is random and thus not known today,
this result only holds when the risk premium term equals the convexity term, λ =
1
2 V (r (1, 2)), as can be seen from equation (7.11).
However, if the risk premium term is higher than the convexity term, λ >
1
V
(r (1, 2)), then r(0, 2) > 5% = r(0, 1). That is, the term structure of inter2
est rates is rising even if market participants do not expect the 1-year rate to increase
between this year and next year. Vice versa, a rising term structure of interest rates
does not necessarily imply an expectation of higher future rates. Indeed, one interesting implication of this fact is that the forward rate is also higher than 5%,
f (0, 1, 2) > 5% = E[r(1, 2)]. That is, the forward rate is higher than the future
rate that is expected by market participants. It follows that observing a high market
forward rate need not imply that market participants are expecting higher rates in the
future. It may well be that they require a high risk premium to hold long-term bonds.
7.3.2
The Expectation Hypothesis
As mentioned in Section 7.3, the expectation hypothesis refers to the theory that the slope
of the term strucure of interest rates only reflects market participants’ expectation of future
interest rates. This hypothesis has been at the center of much research in the past two
decades, and, notwithstanding its intuitive appeal, it has not received much empirical
support. In this section, we link the expectation hypothesis to the model illustrated in the
previous section, and discuss its empirical support.
12 Jensen’s inequality states that given any random variable x, for any convex function f (x), E[f (x)]
> f (E[x]).
258
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
In particular, if
2
(τ − 1)
Vt (r (t + 1, T )) ,
(7.13)
2
then Equation 7.11 implies that the term structure only depends on expected future yields.
Indeed, after substituting in Equation 7.11 the condition in Equation 7.13, setting T = τ −t,
and subtracting on both sides r (t, t + τ ) × (τ − 1) /τ , a little algebra yields the equivalent
expression
λ=
Et [r (t + 1, t + τ ) − r (t, t + τ )] =
1
[r (t, t + τ ) − r (t, t + 1)]
(τ − 1)
(7.14)
That is, the slope of the term structure (on the right-hand side) is related to the expected
change in the yield r (t, t + τ ) between t and t + 1 (on the left-hand side), as postulated by
the expectation hypothesis.
It is important to establish whether the relation in Equation 7.14 is true or not if we want
to understand the forces shaping the yield curve. This understanding is in turn important
for investors to make informed investment decisions, and for monetary authorities to take
correct policy actions. Campbell and Shiller (1991) test this relation by running the
regression
1
[r (t, t + τ ) − r (t, t + 1)] + ε (t + 1)
(τ − 1)
(7.15)
where ε (t + 1) are error terms that are independent of the slope of the term structure.
Collecting time series data on yields, we can compute the time series of the left-hand side
and the right-hand side, and then test the hypothesis that α = 0 and β = 1. Using zero
coupon yield data from 1964 to 2006,13 Panel A of Table 7.4 shows the regression results
for maturities τ = 2, .5. In particular, β is not only drammatically different from 1 for
every maturity, but it is negative, and significantly so.
It is important to understand the meaning of the result in Panel A of Table 7.4. The
negative β implies that a positively sloped term structure predicts a decrease of future
yields, and vice versa. This is the opposite of the expectation hypothesis, and runs against
the basic intuition about the meaning of the yield curve. In the data, a positively sloped
term structure does not predict future higher rates.
This result is important, because not only does it imply that the expectation hypothesis
(7.14) is violated – a high long-term yield spread does not predict higher future rates on
average – but also that the remaining term in Equation 7.11, namely
[r (t + 1, t + τ ) − r (t, t + τ )] = α + β
2
LRPt (τ ) = λ −
(τ − 1)
Vt (r (t + 1, T )) ,
2
(7.16)
must depend on the slope of the term structure. In Equation 7.16, LRPt (τ ) stands for “Log
Risk Premium” from holding a bond with time to maturity τ = T − t, as further discussed
below. Indeed, if Equation 7.13 does not hold, then we can rewrite Equation 7.11 as
Et [r (t + 1, t + τ ) − r (t, t + τ )] =
13 Specifically,
1
[r (t, t + τ ) − r (t, t + 1)] − LRPt (τ )
(τ − 1)
(7.17)
we used the Fama Bliss discount bond data obtained from CRSP.
UNDERSTANDING THE TERM STRUCTURE OF INTEREST RATES
259
Because in Table 7.4 we find that on average, changes in long-term yields are inversely
related to the slope of the term structure, it follows that LRPt (τ ) must be positively related
to the slope of the term structure.
What is the implication of these results? If we observe a strongly sloped term structure,
we should not hastily conclude that the market expects higher future rates. Quite the
opposite, a strongly sloped term structure implies that market participants require a high
risk premium to hold long-term bonds. This high risk premium in turn implies that on
average, we should expect a high capital gain in long-term zero coupon bond in the next
year. A capital gain in zero coupon bonds can only occur through a strong price increase,
which in turn can only occur if its bond yield decreases compared to today. The implication
is then that a strongly sloped term structure predicts lower future yields, on average, as
documented empirically in Table 7.4.
7.3.3
Predicting Excess Returns
The relation between a premium on long-term bonds and the slope of the term structure can
also be seen by examining the return on investments in long-term bonds versus short-term
bonds. In fact, the return in Equation 7.9 can also be rewritten as (see Appendix):
Pz (t + 1, T )
100
− log
= LRPt (τ )
Et log
Pz (t, T )
Pz (t, t + 1)
(7.18)
where LRPt (τ ) is the Log Risk Premium defined in Equation 7.16. From this equation
and Equation 7.13, it follows that the expectation hypothesis implies LRPt (τ ) = 0. Fama
and Bliss (1987) show that the log risk premium is not zero, but it is related to the forward
spread, the difference between the forward rate and the short term spot rate. Denote by
LERt (τ ) = log
Pz (t + 1, t + τ )
Pz (t, t + τ )
− log
100
Pz (t, t + 1)
(7.19)
the log excess return from holding the long-term zero coupon bond with time to maturity
τ over the short-term one year zero coupon bond. Note that log excess return LERt (τ )
defined in Equation 7.19 is the ex-post realized empirical counterpart to the log risk premium
LRPt (τ ) in Equation 7.18: LRPt (τ ) = Et [LERt (τ )]. Fama and Bliss (1987) then run
the following regression:
LERt (τ ) = α + β [f (t, t + τ − 1, t + τ ) − r(t, t + 1)] + ε(t)
(7.20)
The expectation hypothesis has LRPt (τ ) = 0, and therefore α = β = 0. Panel B of
Table 7.4 shows that instead β is significantly different from zero, and indeed positive. This
finding, again, shows that the excess log return is in fact predictable: When the forward
spread is strongly positive, that is, the term structure is positively sloped, on average
investments in long-term bonds generate a higher return compared to short term bonds.
More recently, Cochrane and Piazzesi (2005) have shown that a specific combination of
forward rates successfully predicts excess log returns. The predicting factor is defined by
xt = γ 0 + γ 1 r(t, t + 1) + γ 3 f (t, t + 2, t + 3) + γ 5 f (t, t + 4, t + 5)
260
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
Table 7.4 Predictability
Maturity τ
α
2
3
4
5
-0.01
0.09
0.16
0.17
Maturity τ
2
3
4
5
Maturity τ
2
3
4
5
Panel A: Yield Change Prediction from Slope
se(α)
β
se(β)
0.27
0.24
0.21
0.21
-0.83
-1.23
-1.59
-1.56
0.52
0.62
0.70
0.76
R2
0.03
0.05
0.07
0.06
Panel B: Log Excess Return Prediction from Forward Slope
α
se(α)
β
se(β)
R2
-0.01
-0.19
-0.43
-0.16
0.14
0.15
0.16
0.07
0.27
0.49
0.69
0.93
0.92
1.22
1.43
1.11
0.26
0.34
0.44
0.51
Panel C: Log Excess Return Prediction from Cochrane Piazzesi Factor
β
se(β)
0.47
0.88
1.22
1.43
0.07
0.13
0.19
0.24
Notes: Regression results based on Fama Bliss discount bond data from CRSP.
Sample: 1964 – 2006.
R2
0.30
0.33
0.35
0.32
COPING WITH INFLATION RISK: TREASURY INFLATION-PROTECTED SECURITIES
261
where γ i , i = 0, 1, 3, 5 are estimated from average log excess returns across maturities.14
Cochrane and Piazzesi (2005) run the regression
LERt (τ ) = β × xt + ε(t)
(7.21)
Panel C of Table 7.4 gives the results in the 1964 – 2006 sample. The coefficients are
strongly positive, and the R2 higher than in Panel B, showing that including information
on the whole term structure helps predict bond excess returns.
7.3.4
Conclusion
The expectation hypothesis, the assumption that a positively sloped term structure of interest
rates implies that market participants expect higher future yields, has been largely rejected
by the data. In fact, quite the opposite implication is true: A positively sloped term structure
predicts lower future yields, because it is related to a risk premium that market participants
require to hold long-term bonds. This result is also consistent with the fact that the forward
spread – the difference between the forward rate and the current short term spot rate –
predicts well monthly and annual returns on long term bonds. In short, returns on zero
coupon bonds are predictable by using some predicting factors and this is due to a variation
in risk premia, rather than variation in expectation of future yields.
These empirical findings have an important implication for bond investors. For instance,
if a bond investor interprets a positively sloped term structure as an indication of a future
increase in yields, then he may be led to sell bonds today to avoid capital losses when
interest rates increase. Because the empirical literature has established that in average, a
positively sloped term structure is correlated with lower future yields, the investors should
rather increase the position in bonds.
One important warning, however, is the following: a positively sloped term structure is
associated with a higher risk premium (and this is why it predicts higher future returns).
However, if there is a risk premium, there must be somewhere some risk that induces the
risk premium. That is, holding long-term bonds on the premise that they will yield a higher
return on average does not mean that this return is riskless. In fact, if a long-term bond
has a risk premium, it is quite possible that the market anticipates the possibility of large
capital losses in this long-term bond, and that’s why it is underpriced (or, the yield is high).
What is the risk that a bond holder is facing when he purchases long-term bonds? There
are many. For instance, for the given expectation of future inflation, a higher uncertainty
about the actual level of future inflation increases risk, because if it turns out that inflation
suddenly increases, the Fed will be led to increase the Fed funds rate, pushing down the
price of long term bonds. Such losses can be substantial, as the Orange County case study
discussed in Section 3.7 in Chapter 3 shows.
7.4
COPING WITH INFLATION RISK: TREASURY INFLATION-PROTECTED
SECURITIES
Treasury coupon bonds are in nominal terms, as they pay a sequence of coupons and the
final principal in dollars. Clearly, how much of a good one can buy with the dollar coupons
5
parameters γ are estimated in a first stage regression, in which LER t = 0.25 ×
is regressed on a
τ =2
constant and (r(t, t + 1), f (t, t + 2, t + 3), and f (t, t + 4, t + 5)). The resulting estimates are γ 0 = −3.26,
γ 1 = −1.87, γ 3 = 3.94, and γ 5 = −1.64.
14 The
262
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
and final principal depends on the inflation between the purchase of the bond and the
coupons or principal payments. Over long periods, the difference in purchasing power can
be quite significant. The CPI index, computed monthly by the Bureau of Labor Statistics
(BLS), provides a weighted average of the value of a basket of representative goods that
U.S. consumers purchase.15 The change in the CPI over time measures the realized inflation
during the period.
Consider a household with a monthly income of $10, 000 and let the CPI represent the
price of the basket itself. The amount of the consumption basket that the household can buy
at a given time t1 is Q(t1 ) = $10, 000/CP I(t1 ). For instance, if CP I(t1 ) = 10, then the
household can purchase 1000 units of the basket underlying the CPI index. If the basket was
only made up of $10-hamburgers, the household could purchase Q(t1 ) = 1000 hamburgers
in month t1 . Consider now a later time t2 and assume that the household monthly income
did not change. Because of inflation, however, assume the CP I(t2 ) = 20. Then, the
household could only purchase Q(t2 ) = $10, 000/CP I(t2 ) = 500 hamburgers, a big loss
in consumption, even if the nominal income did not change between the two periods.
The ratio between the quantities the households could purchase, Q(t2 )/Q(t1 ), measures the loss in purchasing power of the dollar between the two dates. This ratio is
given by Q(t2 )/Q(t1 ) = CP I(t1 )/CP I(t2 ). For instance, in the previous example
CP I(t1 )/CP I(t2 ) = 0.5, which means that at t2 the household can purchase only one
half of the goods it could purchase at t1 with the same amount of dollars.
How does the purchasing power change over time? The solid line in Panel A of Figure
7.5 plots the loss in purchasing power over a five year period from 1968 - 2005, that is, the
ratio CP I(t)/CP I(t + 5). For instance, in January 1968 (the first observation), the ratio
was 0.8. This implies that the loss in the value of the dollar between January 1968 and
January 1973 (five years later) meant that in January 1973 consumers could purchase only
80% of what they could purchase in January 1968. Similarly, in 1976 the index was about
0.61, which implies that one dollar in 1981 (five years after 1976) could only purchase 60%
of what it could purchase in 1976.
The dotted line in Panel A shows the ex-ante time value of one dollar five years in
the future, that is, the discount factor Z(t, t + 5). Most often the ex-ante market value
of one dollar in the future Z(t, t + 5) is below the realized loss in value, Z(t, t + 5) <
CP I(t)/CP I(t + 5). This implies that a zero coupon bond at time t is sufficiently cheap
to make up for the ex-post loss in purchasing power of the dollar. However, it also happens
that Z(t, t + 5) > CP I(t)/CP I(t + 5), which has the opposite implication: The ex-post
loss in purchasing power is above the ex-ante value of one dollar in the future. That is, the
price of a zero coupon bond is too high compared to the realized loss in value of the payoff
from the investment. This is called inflation risk.
Indeed, Panel B of Figure 7.5 presents inflation risk from the opposite perspective. This
figure plots the realized inflation between t and t + 5 (the solid line) and the return on a
5-year zero coupon bond made at time t (the dotted line). If the return on a zero coupon
bond is above inflation, then an investment in the zero coupon bond is sufficient to cover the
increase in consumption good prices. However, as it can be seen, in multiple occasions in
the 1970s the return on the zero coupon bond was not sufficient to cover the inflation rate.
15 In
fact, there are several measures of the CPI, which differ in location and type of goods. The one we refer to
here is non-seasonally-adjusted CPI-U, which is the average of the consumption goods in urban cities, which is
the index used for TIPS.
COPING WITH INFLATION RISK: TREASURY INFLATION-PROTECTED SECURITIES
Figure 7.5
263
Ex-Post and Ex-Ante Time Value of Money on a Five Year Horizon
Panel A: Current Value of $1 in Five Years
0.9
Value
0.8
0.7
0.6
Ex−Post
0.5
Ex−Ante
0.4
1970
1975
1980
1985
1990
1995
2000
2005
Panel B: Realized Inflation over 5 Years and Return on a 5−Year Zero Coupon Bond
120
Inflation / Return (%)
Realized Inflation
100
Return on 5−Year Zero
80
60
40
20
0
1970
1975
1980
1985
1990
1995
2000
2005
Data Source: Bureau of Labor Statistics and CRSP.
The ex-post inflation was too high compared to the anticipated value. Investors anticipating
a high inflation rate would require a higher return on a zero coupon bond to cover the loss
in value of the dollar.
Definition 7.1 Inflation risk refers to the loss of purchasing power of the dollar. All assets
that pay fixed amount of dollars in the future are subject to inflation risk.
To cover themselves from inflation risk, investors may require a risk premium on
securities that pay in dollars. This can be seen in Figure 7.5: From Panel A the price
of a zero coupon bond sank in the 1980s, as investors were wary of the high future inflation,
which did not materialize ex post. The zero coupon bond price remained very low until the
end of the 1990s, in fact, even as inflation had decreased substantially. Panel B makes the
same point from a return perspective: The required return on a zero coupon bond over a 5
year period was over 100% in the 1980s to compensate for the fear of high inflation.
264
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
In 1997, the U.S. Treasury introduced Treasury Inflation Protected Securities (TIPS)
to provide investors an investment vehicle that covers against inflation risk. This in turn
could benefit the Treasury itself, as investors in TIPS would not require an additional risk
premium to hold nominal bonds.16 A decade later, about 10% of the total government debt
is in fact in TIPS.
7.4.1 TIPS Mechanics
TIPS are coupon bonds issued with maturities of 5, 10, and 20 years. The coupon rate
of TIPS is a constant fraction of the principal. The principal, however, is not fixed, but it
changes over time in response to inflation. If the CPI increases, then the principal amount
increases proportionally. This implies that the coupon per period increases also with the
CPI, as does the final principal amount. The Treasury publishes index ratios, which are
simply given by the change in the CPI index between the issuance of the TIPS and the
reference CPI reading. The reference CPI reading is not the current CPI, however, but the
average of the CPI value at the beginning of the month of the coupon payment and the CPI
value at the beginning of the previous month. Table 7.5 contains quotes from Treasury
nominal coupon notes and bonds and TIPS on November 26, 2007. In particular, the last
three columns of Panel C report the reference CPI of each of the TIPS, and the “current”
CPI, out of which the index ratio is computed. Given the index ratio, it is possible to
compute the next coupon payment, as it is given by
Coupon payment =
Coupon rate
× 100 × Index ratio
2
The quote in Table 7.5 is for November 26, 2007. The CPI used to compute the index
ratio is not, as noted above, the November CPI (released typically during the third week
of each month by the BLS). Instead, it is the average between the August and September,
2007 CPI reading, which were in fact 207.917 and 208.490, respectively.17 It follows that
an investor in TIPS is subject to a small inflation risk, the inflation that occurs during the
two months between the CPI measure and the actual payment.
7.4.2 Real Bonds and the Real Term Structure of Interest Rates
To understand the valuation of inflation linked securities, it is useful to describe the concept
of “real bonds.” In Chapter 2 we examined the borrowing and lending problem of the
Treasury in terms of dollars. The Treasury borrows some amount of dollars at time t to
return some more in the future T . Although it is intuitive to use dollars as a unit of account
to describe borrowing and lending, as well as the concept of interest rates, this is by no
means unique. Borrowing and lending can occur in any unit, as we know from day-to-day
life. You can borrow a car from a friend and return it filled up with gas. There is no dollar
exchange here, and the implicit “interest rate” depends on the price of gas. Similarly, and
more relevant for finance, gold mine companies often use gold bullion loans to finance
16 On
the other hand, the TIPS program also prevents the government from gaming the investors and cover its
nominal government debt by an inflationary monetary policy.
17 The first day of the month is always equal to the CPI of three months before, in this case the CPI used on
November 1, 2007 = 207.917 = August reading. Because November 26 is 25 days after the first of November, the
calculation is 25/30 × 208.490 + 5/30 × 207.917 = 208.3945.
Table 7.5
Treasury Securities on November 26, 2007
Panel A: Treasury Bills
Maturity
BID
ASK
12/20/2007
2/21/2008
5/22/2008
3.54
3.05
3.24
3.5
3
3.23
Panel B: Nominal Treasury Notes and Bonds
Coupon
Maturity
BID
ASK
3 5/8
4 1/2
3 7/8
4 1/4
5
10/31/2009
5/15/2010
10/31/2012
11/15/2017
5/15/2037
101.25
103.75
102.7188
103.1563
111.7813
101.25
103.7813
102.7188
103.1563
111.8438
Panel C: TIPS
Coupon
Maturity
BID
ASK
Issue Date
Issue Price
First Coupon
Reference CPI
CPI
Index Ratio
2
2 5/8
2 3/8
3 3/8
4/15/2012
7/15/2017
1/15/2027
4/15/2032
104.2188
110.125
109.5313
133.4063
104.2813
110.2188
109.7188
133.625
4/30/2007
7/16/2007
1/31/2007
10/15/2001
102.667
102.722
99.57
98.314
10/15/2007
1/15/2008
7/15/2007
4/15/2002
202.9214
207.2564
201.6645
177.5
208.3945
208.3945
208.3945
208.3945
1.02697
1.00549
1.03337
1.17405
265
Data Source: Bloomberg, Inc.
COPING WITH INFLATION RISK: TREASURY INFLATION-PROTECTED SECURITIES
Coupon
266
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
exploration and mine development: In a gold bullion loan a mining company borrows
some amount of gold at time t with the agreement of returning some more amount of gold
at a later time T , plus possibly gold coupons delivered over time. Once again, there is
no explicit reference to dollars in the transaction, and the effective “dollar” interest rate
depends on the realized price of gold.
Real bonds are bonds that are denominated in units of a good, such as gold, instead of
dollars. Relevant to inflation linked securities are bonds that are denominated in units of
the consumption basket that underlies the CPI index calculation.
Exactly as we did in Chapter 2 when we discussed the discount factor Z(t; T ), we can
now consider the real discount factor Z r eal (t; T ).
Definition 7.2 The real discount factor Z r eal (t; T ) defines the exchange rate between
consumption goods at t versus consumption goods at a later date T .
The quantity Z r eal (t; T ) measures the units of the consumption basket that a consumer
is willing to give up at time t in order to receive one unit of consumption good at time
T . The reasonable behavioral assumption is that to induce somebody to give up some
consumption good today (t), he or she must receive more of it at the later date T . That is,
Z r eal (t; T ) < 1.18
The definition of a real discount factor allows us to then define anything else in terms
of it, exactly as we did in Chapter 2. In fact, all of the concepts described there for the
nominal discount factor Z(t; T ) can be equally defined in terms of the real discount factor
Z r eal (t; T ).
For instance, given a real discount factor, we can compute the real interest rate in the
usual fashion. In the next definition we only consider the continuously compounded real
interest rate. The definition of the real rate at any other compounding frequency can be
obtained as in Chapter 2.
Definition 7.3 The continuously compounded real interest rate can be obtained from the
real discount factor as the solution to the equation
Z r eal (t; T ) = e−r r e a l (t;T )(T −t) × 1
In particular,
rr eal (t; T ) = −
ln Z r eal (t; T )
T −t
(7.22)
(7.23)
The real term structure of interest rates at time t is given by rr eal (t; T ) for various
maturities T .
Similarly, the value (in consumption goods) of a real coupon bond, with maturity T and
coupon rate c is given by
Pcr eal (t; T )
18 For
c × 100
=
2
n
Z r eal (t; Ti ) + 100 × Z r eal (t; T )
i= 1
instance, if somebody borrows your car and gives it back to you after five years, perhaps you are not too
happy, even if it is in the same conditions as it was when you lent it. Likely, you want something else in addition
to the original car, as compensation for the fact that you could not use it for all that time.
COPING WITH INFLATION RISK: TREASURY INFLATION-PROTECTED SECURITIES
7.4.3
267
Real Bonds and TIPS
We can finally see the connection between real bonds and TIPS, and therefore obtain a
pricing formula. It is convenient to start from zero coupon bonds. Suppose an investment
bank purchases a TIPS and strips the coupons from principal, generating a series of zero
coupon bonds. These zero coupon bonds pay an amount that is tied to the CPI. Denoting
by Idx(T ) the CPI adjustment for maturity T (recall, it depends on the CPI two months
earlier), the payoff of a zero coupon TIPS is as follows:
Zero coupon TIPS payoff at T = 100 ×
Idx(T )
Idx(0)
(7.24)
Note that the ratio Idx(T )/Idx(0) represents the increase in the price of the consumption
good between 0 and T (minus two months). Let for simplicity Idx(T ) represent in fact
the cost of purchasing exactly one unit of the consumption basket underlying the CPI. It
follows that 100 × Idx(T ) is the price at T of 100 units of the consumption basket. Given
the real discount factor Z r eal (t; T ) introduced in the previous section, it follows that the
value today (at t) of this payoff is simply
Present value of 100 × Idx(T ) = Z r eal (t; T ) × 100
This present value though is expressed in terms of units of the consumption basket. We
can convert this value to dollars by multiplying it by the current price of the consumption
basket Idx(t). We then obtain
Dollar present value of 100 × Idx(T ) = Z r eal (t; T ) × Idx(t) × 100
Finally, the left-hand side is not exactly equal to the payoff of the zero coupon TIPS. We
must also divide it by the value of the index at time 0, Idx(0), obtaining the pricing formula
Idx(t)
× 100
Idx(0)
(7.25)
In the above computation we are implicitly making the assumption that the time lag between
the final payment T and the index determination does not matter. This assumption simplifies
the computation significantly.
Given the value of zero coupon TIPS, we can compute the value of any coupon bearing
TIPS. For instance, a TIPS value at t, with maturity T , and coupon rate c is given by
n
c × 100
Idx(t)
T IP S
r eal
r eal
×
(t; T ) =
Z
(t; Ti ) + Z
(t; T )
(7.26)
Pc
Idx(0)
2
i= 1
Dollar value of a zero coupon TIPS = PzT I P S (t; T ) = Z r eal (t; T ) ×
7.4.4
Fitting the Real Yield Curve
If we had the value of the real discounts Z r eal (t; T ) we could then price all the TIPS
directly from Equation 7.26. Unfortunately, Z r eal (t; Ti ) are not observable, but they are
embedded in the prices of TIPS. This is also true for the Treasury nominal discount curve
Z(t; T ). However, as explained in Chapter 2 for nominal bonds, we can use the price of
268
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
TIPS to “back out” the discount factors Z r eal (t; Ti ) itself. One limitation compared to
the case of Treasuries is that we do not have available as many bond prices, and therefore
the bootstrapping strategy discussed in Section 2.4.2 is not applicable. However, in the
Appendix of Chapter 2 we reviewed the curve fitting method using a flexible function for
the discount factor, such as the Nelson Siegel model. In this subsection we use the data in
Table 7.5 to illustrate the computation of the real curve.
EXAMPLE 7.3
Recall the curve fitting methodology we employed in Chapter 2, Section 2.9.3.2.
Indeed, from the pricing formula in Equation 7.26, for each TIPS we can compute
the adjusted TIPS price, given by
c × 100
P T I P S (t; T )
PcT I P S (t; T ) = c
=
Idx(t)/Idx(0)
2
n
Z r eal (t; Ti ) + Z r eal (t; T )
(7.27)
i=1
The right-hand side of Equation 7.27 is exactly the same formula used in nominal
bonds in Chapter 2. We can then now use the same steps as in other fitting exercises.
In particular, we compute the adjusted invoice prices, by calculating the accrued
interest to be paid to the seller of the TIPS. Next, we posit a model for the discount
curve Z r eal (0; T ). Here, we use the extended Nelson Siegel model, which states that
the continuously compounded (real) interest rate is given by
1 − e− κ 1
T
rr eal (0, T )
= θ0 + (θ1 + θ2 )
T
κ1
1 − e− κ 2
T
− κT
− θ2 e
1
+ θ3
T
κ2
− e− κ 2
T
(7.28)
There are six parameters to estimate from the quoted prices.19 The solid line in Panel
A of Figure 7.6 plots the term structure of real rates rr eal (0; T ) across maturity. The
real rate is between 1.5% at the low end of the term structure to over 2% at the high
end.
7.4.5
The Relation between Nominal and Real Rates
To conclude this chapter, we could reasonably ask what is the relation between nominal
bonds and inflation indexed bonds? Clearly, the values of these securities cannot be completely independent of each other. For instance, if the real discount Z r eal (t; T ) decreases,
this means that households value future consumption less, and thus want a higher compensation to hold real (or inflation linked) bonds. A nominal bond provides a fixed amount of
dollars in order to purchase the consumption good: However, if such consumption good is
not as valuable in today’s “consumption value,” it is intuitive that the nominal bonds will
have a lower price as well. This discussion can be formalized as follows.
Consider a nominal zero coupon bond at time 0 with maturity T . Its nominal (dollar)
price is
(7.29)
Pz (0, T ) = e−r (0,T )×T × $100
19 From Table 7.5 there are only four quoted prices, but we search for six parameters.
If these were linear equations,
there would be infinite solutions. However, the high non-linearity of the problem makes the problem well defined,
and a unique solution (estimate) can be found. The estimated parameters are θ 0 = 6277.748, θ 1 = −6277.734,
θ 2 = −6288.682, θ 3 = 0.029, κ 1 = 3641.997 and κ 2 = 4.688.
COPING WITH INFLATION RISK: TREASURY INFLATION-PROTECTED SECURITIES
269
Figure 7.6 Real and Nominal Rates on November 26, 2007
Panel A: Real and Nominal Term Structure
Real / Nominal Rates (%)
4.5
Real
Nominal
4
3.5
3
2.5
2
1.5
1
0
2
4
6
8
10
12
Time to Maturity
14
16
18
20
16
18
20
Panel B: Nominal versus Real Rate Spread
3
Spread (%)
2.5
2
1.5
1
0
2
Data Source: Bloomberg.
4
6
8
10
12
Time to Maturity
14
270
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
Having $100 at T will allow us to purchase a quantity $100/Idx(T ) of the consumption
bundle. We consider first the case in which investors have perfect foresight of future
inflation.
7.4.5.1 Nominal and Real Rates under Perfect Foresight Suppose that we
have perfect foresight, so that we know exactly what the CPI index (Idx(T )) is at T . It
follows that the present value of this amount of consumption good can be obtained by
discounting it using the real rate:
Present value of
$100
$100
= e−r r e a l (0,T )×T ×
Idx(T )
Idx(T )
This present value is expressed in terms of consumption good, while we want a dollar price.
We obtain the conversion to dollars by multiplying the present value by the current CPI
index Idx(0):
Idx(0)
$100
= e−r r e a l (0,T )×T ×
Dollar present value of
× $100
(7.30)
Idx(T )
Idx(T )
Comparing this equation with Equation 7.29 we see that in either case we are discounting
$100 to today. Since we are assuming perfect foresight regarding inflation, it follows that
the two discounts are identical to each other, obtaining the relation
e−r (0,T )×T = e−r r e a l (0,T )×T ×
Idx(0)
Idx(T )
(7.31)
Let π be the constant continuously compounded, annualized inflation rate between 0 and
T , that is, π is defined by
Idx(T ) = Idx(0) × eπ ×T
(7.32)
We then obtain from Equation 7.31
r(0, T ) = rr eal (0, T ) + π
Under perfect foresight, the nominal rate equals the real rate plus the (annualized) inflation
rate.
7.4.5.2 Nominal and Real Rates under Uncertain Inflation Because we do not
know the inflation rate between 0 and T (if we did, we would not need TIPS), we need to
modify the analysis in two ways: First, we need to introduce some randomness in future
inflation. Second, we have to take into account the fact that investors in nominal bonds
want to be compensated with a risk premium.
Assume for instance that π in Equation 7.32 has a normal distribution with mean π and
variance σ 2π :
π ∼ N (π, σ 2π )
This implies that the expected loss in purchasing power is
T 2 2
Idx(T )
E
= e−π ×T + 2 σ π
Idx(0)
Second, assume that investors require an (annualized) inflation risk premium κ to hold
a security that pays in dollars, instead of being indexed to inflation. The present value
SUMMARY
271
expression analogous to Equation 7.30 but that takes into account the randomness of
inflation and the inflation risk premium is then given by
$100
−(r r e a l (0,T )+κ)×T Idx(0)
= E e
Dollar present value of expected
× $100
Idx(T )
Idx(T )
=
e−(r r e a l (0,T )+κ)×T × e−π ×T +
T 2
2
σ 2π
× $100
Since this expression must equal to Equation 7.29, it then follows that
r(t, T ) = rr eal (t, T ) + π + κ −
T 2
σ
2 π
(7.33)
That is, the nominal rate equals the real rate, plus the expected inflation and an inflation
risk premium. In addition, there is a (negative) convexity term that appears in the equation,
which is due to the convex relation between the CPI index (Idx(T )) and its growth rate
π. Variation of the real rate then affects the nominal rate, even if the expected inflation
and risk premium are constant. Vice versa, even if the real rate is constant, the nominal
rate may increase or decrease because of variation in expected inflation or the inflation risk
premium κ.
EXAMPLE 7.4
To illustrate the relation between nominal and real rate, consider again Example 7.3.
Fitting the extended Nelson Siegel model to the T-bill, T-notes, and T-bond prices
in Table 7.5 we obtain the nominal spot rate curve r(0; T ) depicted as the dotted
line in Panel A of Figure 7.6. The difference between the two curves provides an
estimate of the combined level of expected inflation π, risk premium κ, and convexity
−(T /2)σ 2π . This quantity is plotted in Panel B of Figure 7.6.
7.5
SUMMARY
In this section we covered the following topics:
1. Basics of monetary policy and the Federal Reserve’s role in the economy: The Fed
has the dual mandate to keep low inflation (stable prices) and low unemployment.
The Fed affects interest rates and thus the return on fixed income instruments through
the determination of its target Federal funds rate and discount rate.
2. Fed funds rate and macro variables: The Federal funds rate moves over time in
relation to inflation and payroll growth. These macro variables help predict future
Fed funds rates.
3. Fed funds futures. Fed funds futures help predict future Fed funds rates. Macro
variables add explanatory power.
4. Expectation hypothesis: The expectation hypothesis states that long-term yields
depend only on market participants’ expectation of future yields. It is strongly
rejected in the data, implying that risk premia and, in fact, time varying risk premia
are a fundamental source of variation in bond yields.
272
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
Table 7.6 Parameter Estimates of Model 1 and Model 2
Model 1
Model 2
α
β1
β2
β3
β4
-0.000196
-0.000112
0.9594
0.1395
0.0731
0.0242
0.0326
0.0062
0.8454
5. Risk premium: A risk premium is the higher average return on an investment over
a given horizon, such as one year, that market participants require to hold longterm bonds over safe short-term bonds, whose return over the investment horizon is
known. The evidence shows that the risk premium is correlated with the slope of the
term structure of interest rates.
6. Treasury Inflation Protected Securities (TIPS): TIPS are Treasury securities whose
principal is indexed to inflation, specifically, the consumer price index (CPI). These
securities offer protection against an increase in inflation.
7. Real rate: The real rate is the rate of interest of an investment net of inflation.
8. Real term structure of interest rates: The relation between yields of real zero coupon
bonds and maturity is known as the real term structure of interest rates. It can be
estimated from TIPS.
7.6 EXERCISES
1. In this chapter we estimated two models to predict the Fed funds rate
rF F (t) = α + β 1 rF F (t − 1) + β 2 X P ay (t − 1) + β 3 X I n f (t − 1)
rF F (t) = α + β 1 rF F (t − 1) + β 2 X P ay (t − 1) + β 3 X I n f (t − 1) + β 4 f F u t (t, t + 1)
Table 7.6 summarizes the estimates in Tables 7.1 and 7.3.
(a) Use the estimates in Table 7.6 to perform a one-month-ahead prediction of the
Fed funds target rate using the data in Table 7.7. That is, for instance, using the
entries on Dec-06, compute the predicted Fed funds rate on Jan-07. Similarly,
using the data on Jan-07 to predict the Fed Funds rate on Feb-07. And so on.
Perform the exercise using both models.
(b) The first column in Table 7.7 provides the actual ex-post Fed Funds target rate.
Plot both models predictions of the Fed funds rate and the actual values. How
close are the estimates?
(c) Compute the sum of squared errors for both models. Based on this calculation,
which model seems to be more accurate?
2. Today is December 12, 2008 and TIPS prices are in Table 7.8.
(a) Use the extended Nelson Siegel model in Equation 7.28 to calculate the real
discount curve and real yield curve.
EXERCISES
Table 7.7
273
Fed Funds Target Prediction
Date
Fed Funds Target
Payroll Growth
Annual Inflation
Fed Funds Futures
Dec-06
Jan-07
Feb-07
Mar-07
Apr-07
May-07
Jun-07
Jul-07
Aug-07
Sep-07
Oct-07
Nov-07
Dec-07
Jan-08
Feb-08
Mar-08
Apr-08
May-08
Jun-08
Jul-08
Aug-08
Sep-08
5.25%
5.25%
5.25%
5.25%
5.25%
5.25%
5.25%
5.25%
5.25%
4.75%
4.75%
4.50%
4.25%
3.50%
3.00%
2.25%
2.25%
2.00%
2.00%
2.00%
2.00%
2.00%
0.08%
0.07%
0.13%
0.06%
0.11%
0.10%
0.07%
0.00%
0.08%
0.12%
0.07%
0.01%
-0.01%
-0.05%
-0.06%
-0.01%
-0.04%
-0.05%
-0.04%
-0.06%
-0.12%
2.08%
2.44%
2.75%
2.57%
2.68%
2.65%
2.37%
1.94%
2.76%
3.54%
4.37%
4.12%
4.40%
4.12%
4.00%
3.88%
4.08%
4.90%
5.52%
5.36%
4.94%
5.24%
5.25%
5.25%
5.25%
5.25%
5.25%
5.25%
5.25%
5.25%
4.66%
4.63%
4.22%
4.16%
2.96%
2.67%
2.17%
2.00%
2.01%
2.01%
2.02%
2.02%
Data Source: Federal Reserve and Bureau of Labor Statistics.
274
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
Table 7.8
TIPS Data on December 12, 2008
Security
Maturity
Price
Coupon
IndexRatio
1
2
3
4
1/15/2014
1/15/2016
7/15/2014
1/15/2026
91 7/32
91
90 1/2
85 25/32
2
2
2
2
1.18406
1.10231
1.16067
1.10231
Source: Bloomberg.
(b) Your estimates for the extended Nelson Siegel model’s parameters should be
close to θ0 = 6278.3013, θ1 = −6278.227,θ 2 = −6289.189, θ3 = −0.18763,
κ1 = 27056.491, and κ2 = 32.190532. Given Equation 7.28, which describes
the application of the extended Nelson Siegel model to real interest rates, can
you see what the short-term interest rate r(0) [i.e., r(0, T ) when T −→ 0] is
by looking only at the parameters?
(c) Using the real term structure, price the a 2% Coupon TIPS with maturity
4/14/2012 and index ratio 1.07817.
(d) On December 12, 2008, the TIPS priced in Part (c) was actually trading at 95
15/16. Is your price close to trading price?
3. Consider the TIPS data in Table 7.8. Assume that you decide to make a portfolio
from Securities 1 and 3 in the table. Specifically, you decide to strip the coupons
from the TIPS. You decide to short the next five coupon payments of Security 1 and
go long the next five coupon payments of Security 3.
(a) Because you are essentially short and long 2% in real terms, is the price of the
total position zero? Explain.
(b) What is the nominal price of this position? Assuming today is December 12,
2008, use the estimates for the extended Nelson Siegel model (Equation 7.28)
given in the previous exercise, i.e., θ0 = 6278.3013, θ1 = −6278.227,θ 2 =
−6289.189, θ3 = −0.18763, κ1 = 27056.491 and κ2 = 32.190532. Do you
have to pay or do you receive money?
(c) The next coupon payment occurs on January 15, 2009, and you are given the
index ratios for these two securities on that date. They are 1.16196 for Security
1 and 1.08173 for Security 3. What will the actual cash flow be at that date?
Do you receive money or do you pay money?
(d) Is this portfolio similar to holding inflation risk or is it more like holding
insurance against inflation risk?
4. Consider the TIPS data in Table 7.8. Assume that you decide to make a costless
portfolio from Securities 1 and 3 in the table. Specifically you decide to strip the
coupons from the TIPS. You decide to short the next five coupon payments of Security
1 and go long the next five coupon payments of Security 3.
CASE STUDY: MONETARY POLICY DURING THE SUBPRIME CRISIS OF 2007 - 2008
275
(a) In order to be costless, what will be the ratio of Security 3 to Security 1 that
you will have to purchase?
(b) What will the next coupon payment be in nominal terms?
(c) What will the coupon payments of this position be in real terms?
(d) What will the price of this position be in real terms? Can you back out the
nominal price from the real price?
5. An important feature of TIPS is that the principal amount cannot go below 100. In
other words, if the index ratio goes under one this parameter is automatically set to
one. This is important in cases when there is deflation (negative inflation), because
in this scenario the value of the principal decreases. Yet given the previously stated
rule, the fall in the reference CPI cannot go below its original value at inception.
Usually there is little or no deflation so index ratios on TIPS tend to accumulate high
values: Even in the event of deflation the value is too high to send it below one. Yet
this isn’t the case for recently issued on-the-run TIPS. Suppose today is January 16,
2002 and you have the TIPS data in Panel A of Table 7.9.
(a) Use the data in Panel A to estimate the extended Nelson Siegel model (see
Equation 7.28) and compute the term structure of real interest rates. The
parameter values you get for the extended Nelson Siegel model should be
similar to the following: θ0 = 6278.3013, θ1 = −6278.271, θ2 = −6289.657,
θ3 = 0.0573662, κ1 = 27056.491, and κ2 = 24.225094.
(b) Consider now the TIPS in Panel B of Table 7.9. You notice that the index ratio
is very close to one: This means that the security has not accumulated much
inflation since it was issued (October 15, 2001). This makes it very easy for it
to go below one if there is a little deflation in the next period. What happens to
this bond if there is deflation? Is the cash from the coupon higher, lower, or the
same as if there wasn’t any inflation?
(c) Is this bond more or less valuable than any of the others in Panel A?
(d) Consider the real term structure of interest rates computed above. Given those
parameters, what is the price of this security?
(e) What is the squared pricing error of this security? Is it very different from the
ones for the other securities in Panel A?
(f) What is the difference in terms of the clean prices?
(g) Does this analysis reflect you answer in Part (c) above?
7.7
CASE STUDY: MONETARY POLICY DURING THE SUBPRIME CRISIS
OF 2007 - 2008
The subprime crisis in 2007 to 2008 offers the occasion to observe the Federal Reserve in
action, as the Fed made use of all of the existing monetary policy tools, and introduced new
ones, in its attempt to avoid allowing the U.S. to enter into a long recession. Table 7.10
lists the key events taking place.20
20 Thanks
to Javier Madrid who put together this case study.
276
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
Table 7.9 TIPS Data on January 16, 2002
Security
Maturity
Price
Coupon
IndexRatio
1
2
3
4
1/15/2011
1/15/2010
4/15/2029
4/15/2028
Panel A: TIPS
100.66
105.81
107.13
102.56
3.50
4.25
3.88
3.63
1.02017
1.05533
1.08006
1.09778
5
4/15/2032
Panel B: On-the-run TIPS
99.88
3.38
1.00031
Source: Bloomberg.
7.7.1 Problems on the Horizon
In mid-2007 the economy seemed to be in good shape, but reports of higher than expected
default rates on loans, specifically in mortgage backed securities (MBS) linked to the
subprime mortgage market were starting to surface. On June 12, 2007, Bear Stearns - an
investment bank - reported a 23% loss on one of its hedge funds (at one time worth $642
million). The High-Grade Structured Credit Strategies Enhanced Leverage Funds were
part of the whole wide range of asset management strategies based on collateralized debt
obligations (CDOs) that Bear Stearns had developed in the past few years. Bear Stearns
was the No. 2 underwriter of mortgage bonds after Lehman Brothers, another investment
bank. Bear Stearns looked to liquidate the fund but found little demand for it. But the
Bear Stearns news was not so noteworthy in an otherwise sound economy, and at its FOMC
meeting on June 28, 2007, the Federal Reserve decided to keep the rates unchanged, as
inflation had been rising and payroll growth was still strong.
In July 2008 some additional bad news hit the market. First, reports came out that in
May, defaults increased 87% when compared to the previous year, especially in the area of
loans in the subprime market. On July 19, 2007, Fed Chairman Ben J. Bernanke testified to
the Senate that losses on securities tied to subprime mortgages could range as high as $100
billion. On July 24, 2007, Countrywide, one of the nation’s largest independent mortgage
lenders, announced that it was forced to take impairment charges due to the rising number of
delinquencies and mortgage defaults. Since 2006, Countrywide was the U.S. top financier
of mortgages, with 20% share of the market. Still, at its scheduled FOMC meeting on
August 7, the Federal Reserve decided to keep rates unchanged because of concerns about
inflation.
Just two days after the Fed decision to maintain rates unchanged, a sequence of events
effectively heralded what many have termed the worst financial crisis since the Great
Depression. First, Countrywide mentioned in an SEC filing that the market problems were
unprecedented disruptions. By this time Countrywide had lost a third of its value (equivalent
to $8.8 billion in market capitalization). Other financials such as Washington Mutual, the
largest U.S. savings and loan, and MGIC Investment, the No. 1 mortgage insurer, faced
similar woes. But the main concern for the Federal Reserve was an unexpected disruption
in the credit market: On August 9, 2007 something unusual happened. Both the overnight
LIBOR and the Fed funds rate had risen sharply over the Fed funds target (see Figure
CASE STUDY: MONETARY POLICY DURING THE SUBPRIME CRISIS OF 2007 - 2008
277
Table 7.10 The Events of the Subprime Crisis: 2007 – 2008
Date
Event
Friday, June 15, 2007
Bear Sterns tries to liquidate holdings of one hedge fund (MBS).
Thursday, June 28, 2007
FOMC meeting. No rate change.
Friday, July 6, 2007
Tuesday, July 17, 2007
NFP go up by 132,000; unemployment is at 4.5% and inflation is at 2.65% (June).
Fed announces investigation on subprime mortgages, due to increase in defaults 87%
year to year.
Tuesday, July 24, 2007
Countrywide reports losses due to high defaults.
Friday, August 3, 2007
NFP go up by 92,000; unemployment is at 4.6% and inflation is at 2.37% (July).
Tuesday, August 7, 2007
FOMC meeting. No rate change.
Thursday, August 9, 2007
Fed injects $24 billion through open market operations.
Friday, August 10, 2007
FOMC unscheduled meeting. Fed injects $35 billion through open market
operations. Fed announces it will take $19 billion in MBS through 3-day repo agreements
at Fed funds rate. Secondary market for mortgages evaporates, pushing Countrywide and
Washington Mutual into deeper trouble.
Thursday, August 16, 2007
FOMC unscheduled meeting. Countrywide takes $11.5 billion credit.
Friday, August 17, 2007
Fed cuts discount rate to 5.75% (down by 50 bps). Additionally, primary credit loans can
now be taken for terms up to 30 days, instead of overnight.
Wednesday, August 22, 2007
Countrywide raises $2 billion by selling 16% stake to Bank of America.
Friday, September 7, 2007
NFP go down by 4,000; unemployment is at 4.6% and inflation is at 1.94% (August).
Tuesday, September 18, 2007
FOMC meeting. Fed cuts Fed funds target rate and discount rate by 50 bps to 4.75% and
5.25%, respectively.
Friday, October 5, 2007
NFP go up by 110,000 and unemployment is at 4.7% and inflation is at 2.76% (September);
Aug revised to + 89,000.
Wednesday, October 31, 2007
FOMC meeting. Fed cuts Fed funds target and discount rate by 25 bps to 4.50% and
5.00%, respectively.
Friday, November 2, 2007
NFP go up by 166,000; unemployment is at 4.7% and inflation is at 3.54%. (October)
Thursday, December 6, 2007
FOMC unscheduled conference call to set up Term Auction Facility and foreign exchange
swap agreement with ECB.
Friday, December 7, 2007
NFP go up by 94,000; unemployment is at 4.7% and inflation is at 4.37% (November)
Monday, December 10, 2007
FOMC meeting. Fed cuts Fed funds target and discount rate by 25 bps to 4.25%, and
4.75%, respectively.
Wednesday, December 12, 2007
Fed starts Term Auction Facility (TAF) designed to provide liquidity to banks.
Fed lends money to banks accepting a wide range of collateral.
Thursday, December 20, 2007
Change in reserve requirements.
Friday, January 4, 2008
NFP go up by 18,000; unemployment rate is at 5.0% and inflation is at 4.12% (December).
Wednesday, January 9, 2008
FOMC unscheduled conference call discussing possible risks of inflation, and necesity of
cutting interest rates further and its timing.
Friday, January 11, 2008
Countrywide acquired by Bank of America for $4.1 billion.
Tuesday, January 22, 2008
Fed cuts Fed funds target rate and discount rate by 75 bps to 3.50%, and
4.00%, respectively.
Tuesday, January 29, 2008
FOMC meeting. Fed cuts Fed funds target rate and discount rate by 50 bps to 3.00% and
3.50%, respectively.
Friday, February 1, 2008
NFP go down by 17,000; unemployment is at 4.9% and inflation is at 4.40% (January).
278
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
Wednesday, February 27, 2008
Fed Chairman proposes a core inflation target between 1.5% and 2% in a report to Congress.
Friday, March 7, 2008
NFP go down by 63,000; unemployment is at 4.8% and inflation is at 4.12% (February).
Friday, March 7, 2008
Fed boosts size of TAF to $100 billion.
Tuesday, March 11, 2008
Fed expands securities lending program, under the Term Securities Lending Facility (TSLF),
which lends up to $200 billion of Treasury securities to primary dealers for up to 28 days
(instead of overnight) by pledge of other securities including MBS.
Sunday, March 16, 2008
Bear Stearns collapses. JP Morgan Chase agrees to acquire it for $2/share.
Sunday, March 16, 2008
Fed cuts discount rate to 3.25%. (down by 25 bps)
Sunday, March 16, 2008
Fed starts the Primary Dealer Lending Facility (PDLF), providing funding to primary dealers
in exchange for a specified range of collateral
Monday, March 17, 2008
FOMC meeting. Fed cuts Fed funds target rate and discount rate by 75 bps to 2.25% and
2.50%, respectively
Friday, April 4, 2008
NFP go down by 80,000; unemployment is at 5.1% and inflation is at 4.00% (March).
Wednesday, April 30, 2008
FOMC meeting. Fed cuts Fed funds target rate and discount rate by 25 bps to 2.00% and
2.25%, respectively.
Friday, May 2, 2008
NFP go down by 20,000; unemployment is at 5.0% and inflation is at 3.88% (April).
Friday, June 6, 2008
NFP go down by 49,000; unemployment is at 5.5% and inflation is at 4.08% (May).
Wednesday, June 25, 2008
FOMC meeting, no rate change due to concerns on inflation.
Thursday, July 3, 2008
NFP go down by 62,000; unemployment is at 5.5% and inflation is at 4.90% (June).
Thursday, July 24, 2008
FOMC unscheduled conference call in which TSLF is extended to 01/30/09 and allows
options to be offered (up to $50 billion).
Friday, August 1, 2008
NFP go down by 51,000; unemployment is at 5.7% and inflation is at 5.52% (July).
Tuesday, August 5, 2008
FOMC meeting, no rate change due to concerns on inflation.
Friday, September 5, 2008
NFP go down by 84,000; unemployment is at 6.1% and inflation is at 5.36% (August).
Sunday, September 7, 2008
Fannie Mae and Freddie Mac placed under government ’conservatorship’.
Sunday, September 14, 2008
Merrill Lynch merges into Bank of America.
Monday, September 15, 2008
Lehman Brothers defaults on its commercial paper as it goes bankrupt ($613 billion debt).
Tuesday, September 16, 2008
FOMC meeting, no rate change due to concerns on inflation.
Tuesday, September 16, 2008
Government bails out American International Group (AIG) with a $85 billion package.
Wednesday, September 17, 2008
Lehman Brothers bought separetly by Barclays and Nomura Bank.
Friday, September 19, 2008
Fed starts Asset-Backed Commercial Paper Money Market Mutual Funds Liquidity Facility
which provides liquidity to banks holding asset backed securities.
Friday, September 26, 2008
Washington Mutual is acquired by JP Morgan Chase.
Wednesday, October 1, 2008
Federal Reserve starts to pay Interest on Required Reserve Balances and Excess Balances.
Friday, October 3, 2008
NFP go down by 159,000; unemployment is at 6.1% and inflation is at 4.94%(September).
Friday, October 3, 2008
Wachovia merges into Wells Fargo Bank.
Tuesday, October 7, 2008
Fed announces creation of the Commercial Paper Funding Facility (CPFF)
Wednesday, October 8, 2008
FOMC unscheduled meeting. Fed cuts Fed funds target rate and discount rate by 50 bps to
1.50% and 1.75%, respectively.
Tuesday, October 21, 2008
Fed announces creation of the Money Market Investor Funding Facility (MMIFF)
Wednesday, October 22, 2008
Adjustment of formula for interest on required reserve balances
Sources: The Wall Street Journal, Financial Times, Bureau of Labor Statistics.
CASE STUDY: MONETARY POLICY DURING THE SUBPRIME CRISIS OF 2007 - 2008
Figure 7.7
279
Borrowing Rates during the Subprime Crisis
0.07
0.06
First Liquidity Injection →
0.05
Rate
0.04
0.03
0.02
0.01
0
Overnight LIBOR
Fed Rate
Fed Target Rate
Discount Rate
02/28/07
06/26/07
10/22/07
Time
02/20/08
06/17/08
10/10/08
7.7),with the increase in the LIBOR being particularly dramatic. This prompted the Fed to
intervene to return rates to the target.
7.7.1.1 Open Market Operations The Fed faced the problem that rates were going
beyond the target, which meant that demand for funds was exceeding their supply. The
Fed decided to inject $24 billion on Thursday, August 9 and another $35 billion on August
10, 2007 through open market operations. The specific transaction used to inject this
liquidity was repurchase agreements (see Chapter 1), and was performed, as usual, through
designated institutions called primary dealers. In particular, on August 9, 2007 the Federal
Reserve Bank of New York received $34.7 billion in bids submitted for 14-day repos on
Treasuries. Of these $4 billion were accepted with a 5.18% rate (weighted average). So
for every $100 of Treasuries the seller would have to pay $0.39 in interest after 14 days.
7.7.1.2 Open Market Operations with Mortgage Backed Securities Also, in
addition to the $35 billion in funds made available by the Fed in August 10, $19 billion
was made available through a 3-day MBS repo agreement which meant that instead of
Treasuries the Fed was willing to take MBS as collateral for the repo agreement. The
MBS given had to be high-investment grade (minimum default risk). This allowed primary
dealers that were heavy in MBS (such as Lehman Brothers and Bear Stearns) to obtain extra
financing. On August 9, 2007 the Federal Reserve Bank of New York received $36.5 billion
in bids submitted for 14-day repos on Treasuries. Of these $5.93 billion were accepted with
a 5.33% rate (weighted average). Interest payment per $100 of MBS to the seller would be
$0.40 to be paid in 14 days.
280
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
7.7.2 August 17, 2007: Fed Lowers the Discount Rate
August 16, 2007 was another bad day. Markets were shaken on news that Countrywide
took a $11.5 billion line of credit to maintain liquidity, which further wore on the subprime
market and increased the actual size of the crisis. Both the Dow Jones Industrial and the
NASDAQ fell sharply. As a response to these events, and worries of a widespread credit
crisis, the Federal Reserve lowered the rate offered as the discount rate. The discount rate is
the rate at which depository institutions can borrow funds from the Fed. Originally thought
to be a measure of last resort in case a bank couldn’t fulfill its reserve requirements in a day,
the Fed decided to lower the discount rate in an effort to increase liquidity in the market.
The rate was lowered from 6.25% to 5.75%. Historically, the discount rate was usually
moved in unison with the Fed funds target rate, always maintaining the same spread. Up to
this date the spread was one percentage point. Yet by lowering the discount rate unilaterally
by 50 basis points the spread was halved.
In addition, the Federal Reserve increased the type of collateral accepted for loans
within the discount window facility, and the lending terms for primary credit (i.e. to sound
institutions) extended from overnight to up to thirty days. One important point to note is
that only FDIC approved depository institutions have access to the discount window, but
not all primary dealers are in fact depositary institutions. We return to this point below.
7.7.2.1 The SOMA Securities Lending Program An additional policy tool that
the Federal Reserve has is the System Open Market Account (SOMA) Securities Lending
Program, in which a bank can bid for Treasury securities from the SOMA portfolio to be
held overnight. Each day, at noon, the New York Fed holds an auction in which primary
dealers bid, with the fee to be paid, for the securities. Loans are given on a bond-versusbond lending scheme, in which the primary dealer must offer Treasury bills, notes, bonds
and inflation-indexed securities (instead of cash) as collateral. Because Treasuries are used
in repo agreements, there is an incentive to take advantage of this program to trade on
specials. To limit these trades, the New York Fed normally imposes a minimum bid on the
lending fee. On August 21, the minimum lending fee was reduced to 50 basis points.
7.7.3 September - December 2007: The Fed Decreases Rates and Starts
TAF
As the credit crisis worsened, the Federal Reserve decreased the Fed funds target rate and
the discount rate at its FOMC meetings on September 18, October 31, and December 12
(see Figure 7.7), to 4.25% and 4.75%, respectively. At the December FOMC meeting, the
Federal Reserve started a new policy, TAF, discussed next.
7.7.3.1 Term Auction Facility (TAF) Under this policy eligible depository institutions can bid for a given amount of funds auctioned by the Fed. The loans are collateralized.
Requirements for entering the auction are the same as those for primary credit loans through
the discount window. The auction works in the following way. Suppose a bank having
primary credit status decides to participate in the program. On December 17, 2007, the
day of the first auction, it enters the auction with an amount and a bid in terms of interest
rate. The amount of the bid cannot be more than 10% of the funds offered and the interest
rate on the bid is subject to a prespecified lower limit. In this case the minimum rate to be
offered was 4.17% and the funds offered were $20 billion in a 28-day loan. Because the
CASE STUDY: MONETARY POLICY DURING THE SUBPRIME CRISIS OF 2007 - 2008
281
funds have a 28-day maturity, they follow the same collateralization requirements as for the
discount window. So the bank could pledge its $10 million in T-bills for $9.8 million for
4.7%, for instance. Once the bids close, the Fed awards the first set of funds to the highest
bidder and then goes down the list awarding the funds to those with the next highest bids.
The rate at which the last participant is awarded funds is called the stop-out rate (or lowest
successful bid). On this date the stop-out rate was 4.65%, which means that the bank in
our example got the loan.
7.7.4
January 2008: The Fed Cuts the Fed Funds Target and Discount
Rates
In January, 2008 the Federal Reserve slashed both its reference interest rates. Worried
about a worsening economic outlook, the Federal Reserve dropped both the Fed funds
target rate and discount rate by 75 basis points on January 22 at an unscheduled meeting,
and then again by an additional 50 basis points just a few days later, at its scheduled FOMC
meeting. By January 29, the Fed funds target rate and discount rate dropped to 3.00% and
3.50%, respectively.
7.7.5
March 2008: Bearn Stearns Collapses and the Fed Bolsters Liquidity
Support to Primary Dealers
On March 11, 2008, the Fed announced that the SOMA Securities Lending Program would
be expanded both in length to 28 days and in the range of collateral allowed, including
investment grade corporate securities, municipal securities, mortgage-backed securities,
and asset-backed securities. Essentially, under the expanded program, primary dealers
could swap riskier securities, such as MBS, for safer securities, such as Treasuries.
On Sunday, March 16, Bearn Stearns collapsed. JP Morgan Chase agreed to acquire
Bearn Stearns for $2/share. On the same date, the Fed further closed the gap between the
discount rate and the Fed funds target rate, as the discount rate was lowered by 25 basis
points to 3.25%. This narrowed the gap to only 25 basis points. Subsequently, the Fed
funds target rate and discount rate were decreased again on March 17 (by 75 basis points)
and on April 30 (by 25 basis points) to reach 2% and 2.25%, respectively. These would be
the last interest rate cuts for a while because of concerns about inflation picking up. The
Fed did however announce additional programs instead.
7.7.5.1 Primary Dealer Credit Facility (PDCF) On March 16, right after the Bearn
Stearns collapse, the Fed started the Primary Dealer Credit Facility. This program allowed
primary dealers to obtain overnight loans from the Federal Reserve Bank of New York.
Institutions could borrow up to whatever their margin-adjusted collateral would allow. The
rate charged for the overnight loan was the same as the one for primary credit institutions
in the discount window. In fact, in terms of the collateral allowed, the rates charged and the
limits on borrowing, this program is identical to the discount window. The key difference
is that primary dealers, which do not have access to the discount window, are allowed
to participate in this program instead. Nonetheless, the Federal Reserve established an
additional fee for primary institutions that access this facility on more than 30 business
days out of 120 business days.
282
INFLATION, MONETARY POLICY, AND THE FEDERAL FUNDS RATE
7.7.6 September – October 2008: Fannie Mae, Freddie Mac, Lehman
Brothers, and AIG Collapse
During September, 2008 a number of major events took place: First, on September 7, Fannie
Mae and Freddie Mac, the two giant government-sposored agencies and key players in the
mortgage backed securities market, were placed under conservatorship, as they were finding
it difficult to meet their financial obbligations. Second, on September 15, Lehman Brothers
declared bankruptcy. Right afterward, the government bailed out American International
Group (AIG) with an $85 billion package. As the interbanking market reacted badly to the
news (see Figure 7.7), the Federal Reserve started yet another program to support liquidity,
discussed next.
7.7.6.1 Asset-Backed Commercial Paper (ABCP) Money Market Mutual Funds
(MMMF) Liquidity Facility (AMLF) This program allows U.S. depository institutions
and bank holding companies to finance their purchases of high-quality asset-backed commercial paper (ABCP) from money market mutual funds (MMMF). The AMLF is administered by the Federal Reserve Bank of Boston, and allows loans at the primary credit
discount rate subject that these funds are used to meet responsibilities.
7.7.6.2 Interest on Required Reserve Balances and Excess Balances In
2006, Congress passed the Financial Services Regulatory Relief Act, which authorized the
Fed to pay interest on balances held by or on behalf of depository institutions at Federal
Reserve banks effective October 1, 2011. This effective date was advanced to October 1,
2008 by the Emergency Economic Stabilization Act of 2008. As pointed out earlier, holding
reserves, either required or excess, at a Federal Reserve bank implies an opportunity cost
(effectively a tax) on these funds, as they could be gaining interest.
Interest payments on required reserves are obtained by averaging the Fed funds target
rate over the maintenance period (usually calculated weekly) minus a spread. Interest
on excess reserves is computed by taking the minimum Fed funds target rate over the
maintenance period minus a spread. An example is given in Table 7.11
7.8 APPENDIX: DERIVATION OF EXPECTED RETURN RELATION
Take logs on both sides of Equation 7.12 to find
log [Et [Pz (t + 1, T )]] − log [Pz (t, T )] = log (100) − log [Pz (t, t + 1)] + κ
(7.34)
From the assumption about normally distributed yield
2
log [Et [Pz (t + 1, T )]] = −Et [r (t + 1, T )] × (τ − 1) +
Note now that
Et [log (Pz (t + 1, T ))]
(τ − 1)
V [r (t + 1, T )]
2
= Et log e−r (t+1,T )(τ −1)
= −Et [r (t + 1, T )] × (τ − 1)
2
=
log [Et [Pz (t + 1, T )]] −
(τ − 1)
Vt [r (t + 1, T )]
2
APPENDIX: DERIVATION OF EXPECTED RETURN RELATION
283
Table 7.11 Example of Interest Paid on Reserves at the Federal Reserve Bank
Maintenance Period
(Ending On)
October 29, 2008
October 22, 2008
October 15, 2008
Maintenance Period
(Ending On)
October 29, 2008
October 22, 2008
October 15, 2008
Panel A: Interest Paid on Required Reserve Balances
Rate
Average Target Fed Funds
Rate During Maintentance Period
1.33%
1.40%
1.40%
1.43%
1.50%
1.50%
Panel B: Interest Paid on Excess Reserve Balances
Rate
Minimum Target Fed Funds
Rate During Maintenance Period
0.65%
0.75%
0.75%
1.00%
1.50%
1.50%
Spread
0.10%
0.10%
0.10%
Spread
0.35%
0.75%
0.75%
Substitute, to find that Equation 7.34 can be rewritten as
2
Et [log (Pz (t + 1, T ))]−log [Pz (t, T )]+
(τ − 1)
Vt [r (t + 1, T )] = log (100)−log [Pz (t, t + 1)]+κ
2
or
2
Pz (t + 1, T )
100
(τ − 1)
Et log
Vt [r (t + 1, T )]
− log
=κ−
Pz (t, T )
Pz (t, t + 1)
2
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CHAPTER 8
BASICS OF RESIDENTIAL MORTGAGE
BACKED SECURITIES
The residential mortgage backed securities market reached a market size of about $8.9
trillion by the end of 2008. This is almost $3 trillion larger than the marketable U.S.
Treasury debt at the same time. Clearly related to the real estate market, it is no surprise
that the size of this market has been growing steadily over the years. The market for
mortgage backed securities serves the important role of transferring risks from those who
have it, the small banks, savings and loans, and so forth, to those who are better able to
bear it, namely, investors. The latter are more diversified and thus potentially in a better
position to bear the risks of lending money to individuals. The process through which
mortgages to individuals are sold to others is called securitization. While we expand our
discussion on residential mortgage backed securities, a very similar procedure is followed
to securitize almost any type of security, from credit cards, to car loans, from commercial
loans to corporate bonds.
8.1
SECURITIZATION
Before we delve into a discussion of the market for residential mortgage backed securities,
let’s take a closer look at the concept of securitization. The essential idea is in fact simple:
Some institutions hold in their assets investments that are too risky for them, and they would
like to sell them to investors who are better able to bear their risk. Since the assets per se
are too concentrated and not sufficiently diversified, it is hard for a financial institution to
sell individual ones to anyone for a reasonable price. A solution is the following: Several
285
286
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
Figure 8.1 The Securitization Process
Originator 1
Originator 2
···
Originator n − 1
Originator n
Asset Pooling
↓
Special Purpose Vehicle
↓
Asset Backed Securities
↓
Investors
Table 8.1
Examples of Securitized Products
Security’s Name
Collateral Assets
Residential Mortgage Backed Securities (RMBS):
Commercial Mortgage Backed Securities (CMBS):
Assets Backed Securities (ABS):
Collateralized Debt Obligations (CDO):
Residential mortgages with similar characteristics
Commercial mortgages with similar characteristics
Receivables, such as auto loans, credit cards and so on
Investment and high yield corporate bonds, other
structured products, credit default swaps
Corporate loans
Collateralized Loan Obligations (CLO):
institutions may get together, pool similar assets in a portfolio to diversify away the risk
embedded in each asset, and then sell the portfolio to investors. In order to pool the assets,
individual financial institutions can create a standalone firm, called a special purpose vehicle
(SPV), which formally buys the assets by raising the necessary capital from investors. In
the case in which these assets are mortgages, the securities issued by the SPV to investors
are called residential mortgage backed securities (RMBS).
Figure 8.1 describes the securitization process. In this figure, the originator is the
financial institution that wants to pool some assets to sell separately. The issuer is the party
that purchases the assets from the originators. As mentioned, the originator often creates
a SPV to make the pool bankruptcy remote. The idea here is to separate the SPV assets
– the collateral backing the securities – from the balance sheet of the issuer. Trustees are
typically also appointed to ensure that the SPV in fact delivers on its contractual obligations.
There are other third parties involved as well, that are not represented in Figure 8.1. For
instance, there are mortgage servicers who are responsible for collecting the payments from
the homeowners and sending them to the investors, and other third parties who provide
additional credit guarantee to investors.
Securitization occurs in the residential mortgage market, in which individual savings
and loans, thrifts, and other banks pool the mortgage loans on their assets, and sell them to
SPVs in exchange for cash. Originators may also sell their mortgage loans to other issuers,
such as Freddie Mac or Fannie Mae, in exchange for residential mortgage backed securities
(instead of cash), which can then be sold in the secondary market or kept on the banks’
assets. Not only residential mortgages are securitized, but also a large spectrum of other
assets, as indicated in Table 8.1
SECURITIZATION
8.1.1
287
The Main Players in the RMBS Market
There are two types of RMBS: Agency MBS and non-agency MBS. Agency MBS are those
ones in which government agencies are involved. The major players involved in residential
mortgage backed securities are:
1. Ginnie Mae: Government National Mortgage Association (GNMA). Formed by
the U.S. Congress in 1968, Ginnie Mae is a wholly-owned government corporation
within the U.S. Department of Housing and Urban Development. In 1970, Ginnie
Mae developed and guaranteed the first mortgage backed security. Ginnie Mae’s
main function is to guarantee the timely payments of RMBS backed by loans made
through the Federal Housing Administration (FHA) program, the Office of Public
and Indian Housing (PIH) program, and the Department of Veteran Affairs (VA)
Home Loan program. Ginnie Mae does not make or purchase loans, nor does it buy,
sell, or issue securities. Instead, it only guarantees MBS that are issued by approved
private lending institutions, which pool loans and issue RMBS.1
2. Fannie Mae: Federal National Mortgage Association (FNMA). Initially created
in 1938 as a government agency, it changed in 1968 into a shareholder-owned
company, although with a Federal charter, until the credit crisis of 2007–2009 forced
the U.S. Government to place Fannie Mae in conservatorship (that is, an effective
nationalization). As does Ginnie Mae, Fannie Mae provides credit guarantees on
mortgage loans that are securitized through Fannie Mae. As opposed to Ginnie
Mae, Fannie Mae also mantains a large mortgage portfolio and issues debt to finance
its portfolio. By directly operating in the secondary market, Fannie Mae provides
liquidity in the mortgage market, which in turn allows banks to grant mortgages to
individual homeowners at more convenient rates than otherwise possible. Fannie
Mae issued its first mortgage backed security in 1981. Since then, Fannie Mae has
become one of the largest agency issuers of MBS.2
3. Freddie Mac: Federal Home Loan Mortgage Corporation (FHLMC). Also a stockholderowned institution until the credit crisis of 2007 – 2008, Freddie Mac was chartered
by the government in 1970 in order to stabilize U.S. residential mortgage markets
and expand opportunities for homeownership and affordable rental housing. Freddie
Mac follows the same business model as Fannie Mae.3
Mortgage backed securities guaranteed by Ginnie Mae are default free securities, as
they have the explicit backing of the U.S. government. Mortgage backed securities issued
or guaranteed by Fannie Mae and Freddie Mac have also been considered quite safe
investments, in terms of default, as there has been a general market perception that in case
of financial trouble, the U.S. government would step in and rescue the two agencies. In
retrospect, such a belief was actually accurate, as the U.S. governement did in fact rescue
the two agencies when they faced financial troubles during the credit crisis of 2007 – 2008.
These two institutions are very large: As of December, 2007 Fannie Mae had about $882
billion of assets, while Freddie Mac had about $794 billion. Together, these two agencies
1 Source:
Ginnie Mae Annual Report, 2006.
An Introduction to Fannie Mae, 2007.
3 Source: Freddie Mac Annual Report, 2006.
2 Source:
288
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
guarantee around $5 trillion of mortgages, which is slightly more than half of the market
of residential mortgage backed securities.
8.1.2 Private Labels and the 2007 - 2009 Credit Crisis
In addition to the three government institutions, residential mortgage backed securities are
also issued by other institutions. Table 8.2 shows the issuance of mortgage backed securities
between 1996 and 2008. Although the largest share of RMBS is issued by governmentsponsored agencies, nonagency mortgage backed securities issuance share increased over
time up to 2007. This large increase in the private label market parallels the acceleration
in U.S. house prices that occurred from 2000 to 2006, as shown in Figure 8.2. Part of the
reason for the increase in the private label markets is that government-sponsored agencies
have restrictions on the type of mortgages they can take on. For instance, they can only
securitize so-called conventional mortgages, that is, with principal amount below a given
cutoff ($417,000 in 2008), with a loan-to-value ratio of at most 80%. The increase in house
prices increased the demand for jumbo mortgages, i.e., those whose principal amount is
above the cutoff discussed earlier, providing an incentive for other private players to enter the
securitization business. In addition, more and more households requested mortgages with
a loan-to-value ratio higher than 80%, which cannot be securitized through the government
sponsored agencies. The private label market filled the gap, assuming some risks, of course,
as the value of the collateral (the house price) was providing a lower cushion against the
probability of default.
The downturn in house prices in 2007 and 2008, and the subsequent financial crisis (see
the discussion in Section 7.7 of Chapter 7), made investors wary of purchasing mortgage
backed securities that are not backed by the full faith of the U.S. government. Because of
the lack of investors willing to purchase nonagency MBS, in 2008 nonagency MBS issuance
dropped dramatically, as shown in Table 8.2. In fact, a month-by-month breakdown shows
that starting in September of 2008, the issuance of MBS by institutions other than the three
U.S. agencies has been essentially zero. This lack of demand for nonagency MBS directly
affects the market for nonconventional mortgage loans, that is, those with large principal
(jumbo loans) or with a high loan-to-value ratios.
The securitization market is a vital part of the U.S. credit system. As the mortgage
backed securities market grew to an almost $9 trillion market, banks became accustomed
to a new business model according to which they would originate mortgage loans with the
explicit intention to sell them in the securitized market. If the securitization market were to
collapse – in the sense that there would no longer be private investors willing to purchase the
pools of mortgages – so would the mortgage lending market itself, as banks would not want
to keep the mortgage risks in their portfolio. The U.S. Treasury and the Federal Reserve
are very much aware of this problem, as some of their policies in 2008 were directed to
boost the securitization market. For instance, on November 25, 2008 the Federal Reserve
announced “a program to purchase the direct obligations of housing-related governmentsponsored enterprises (GSEs)– Fannie Mae, Freddie Mac, and the Federal Home Loan
Banks – and mortgage backed securities (MBS) backed by Fannie Mae, Freddie Mac, and
Ginnie Mae” (see Federal Reserve Board Press Release, November 25, 2008). Why only
agency debt and MBS? Under its mandate, the Federal Reserve cannot directly purchase
risky debt. In order to do so, the U.S. Treasury has to be involved. Indeed, in the fall of
2008, the U.S. Treasury annouced the Trouble Asset Relief Program (TARP), a $700 billion
SECURITIZATION
289
9
220
8
200
7
180
6
160
5
Mortgage−Backed
Securities Market
140
4
120
Case−Shiller House
Price Index
3
100
2
80
1
60
1985
1990
1995
2000
2005
Mortgage−Backed Securities (trillions of dollars)
Case−Shiller Composite−10 Index
Figure 8.2 House Prices and the Mortgage Backed Securities Market
240
0
2010
Source: SIFMA and Standard & Poor / Case-Shiller.
program aimed at purchasing, directly or indirectly, mortgage backed securities and other
securitized products from banks. A number of additional initiatives aimed at helping the
ailing securitization market have since been announced.
8.1.3
Default Risk and Prepayment in Agency RMBSs
The credit crisis of 2007 – 2009 is characterized by an unusually large number of homeowners’ defaults, meaning that homeowners stopped paying the monthly mortgage coupons.
Because of defaults, a large number of houses went into forclosure, forcing banks to sell
them at fire-sale prices. The Case Shiller index in Figure 8.2 includes foreclosed home
prices, which are partly responsible for the substantial decrease in housing prices in 2008.
To understand the pricing of residential mortgage backed securities, however, we have
to understand the type of risk that an investor in RMBS is facing, and here we need
to distinguish between agency RMBS (the vast majority of the market) and nonagency
RMBS. An investor who purchased agency RMBS is not exposed directly to the credit
risk of the mortgages in the RMBS pool. The reason is that the agencies insure investors
against the default of individual mortgages: In case of default, the agency steps in and
repays the face value of the mortgage to the investors. The risk for RMBS investors is then
to receive the money invested sooner than they expected. For instance, investors who in
2006 invested in 30-year RMBS would have an horizon of several years, as the underlying
pool is expected to make payments for many years. A wave of defaults, however, would
imply a large cash flow back to the investors, as the agencies step in to repay the mortgages.
Prepayments – receiving cash flows too early compared to the expected life of the mortgage
– are the most interesting characteristics of RMBS. The remaining part of this chapter is
devoted to studying this phenomenon, and its pricing and risk implications.
290
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
Table 8.2
Mortgage Related Issuance
Year
Agency
Nonagency
Total
Agency Share
Nonagency Share
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
440.7
535.0
952.0
884.9
582.3
1,454.8
1,985.3
2,725.8
1,375.2
1,321.0
1,214.7
1,372.2
1,299.2
51.9
69.4
191.9
140.5
101.7
218.8
288.5
440.6
532.7
901.2
917.4
773.9
40.5
492.6
604.4
1,143.9
1,025.4
684.0
1,673.6
2,273.8
3,166.4
1,907.9
2,222.2
2,132.1
2,146.1
1,339.7
89.46%
88.52%
83.22%
86.30%
85.13%
86.92%
87.31%
86.09%
72.08%
59.45%
56.97%
63.94%
96.98%
10.54%
11.48%
16.78%
13.70%
14.87%
13.08%
12.69%
13.91%
27.92%
40.55%
43.03%
36.06%
3.02%
Notes: Agency issuance includes GNMA, FNMA, and FHLMC mortgage backed
securities and CMOs. Nonagency issuance includes both
private-label MBS and CMOs. Quantities are $ billions.
Souce: SIFMA. Government-Sponsored Enterprises, Thomson Financial, Bloomberg
8.2 MORTGAGES AND THE PREPAYMENT OPTION
Before further discussing the mortgage backed securities market, it is important to review
some basic facts about standard fixed rate mortgages. Consider for instance a 30-year
fixed-rate mortgage, with mortgage rate of rm
12 . The subscript denotes the frequency of
compounding: Because mortgage coupons are paid at the monthly frequency, the compounding frequency is n = 12. The superscript m denotes that it is a mortgage rate, which
is different from the Treasury rate, as detailed below. Suppose that L is the amount of the
mortgage lent from the bank to the homeowner. According to the standard present value
relation, the periodic coupon must then satisfy
30×12
L=
i= 1
C
rm
12
12
1+
i
(8.1)
It is convenient to define the following constant
A=
Because we can then rewrite L =
8.1 is given by
30×12
i= 1
C=
1
1+
rm
12
12
(8.2)
C × Ai , we obtain that the coupon in Equation
L
30×12
i= 1
Ai
(8.3)
It is important to note an important distinction between a mortgage and a regular bond: In a
regular bond the periodic coupon comprises only the interest on the bond’s principal, while
MORTGAGES AND THE PREPAYMENT OPTION
291
the principal itself is repaid only at maturity. In a mortgage, instead, the principal is repaid
during the life of the mortgage together with the interest. Indeed, the coupon C contains
two components: One component is the interest payments and the other component is the
principal repayment. The fractions of the total coupon C that is related to interest payment
and principal repayment vary over time. The reason is that the interest amount paid is
determined by the amount of outstanding principal, which declines over time as principal
payments occurs. The larger the outstanding principal, the larger the amount of interest
that has to be paid, and thus the larger is the fraction of interest payment in the coupon.
More specifically, the interest and the principal paid at any time t are given by:
Interest paid at t =
rm
12
× Lt
12
= C − It
=
It
Principal paid at t = Lpaid
t
(8.4)
(8.5)
The amount of principal declines over time, as the principal remaining next month equals
the amount of principal this month minus what is paid during this month:
Lt+ 1 = Lt − Lpaid
t
(8.6)
Panel A of Figure 8.3 plots the scheduled principal outstanding Lt over time, for a
30-year, 6%, $300,000 mortgage. Panel B plots the interest rate and the principal paid
component of the monthly coupon, which is about C = $1797.7.
Given the coupon payments C’s to be made over the life of the mortgage, the mortgage
itself can be considered as any other bond. Indeed, the mortgage rate rm
12 is simply the
internal rate of return of the bond (see Section 2.4.3). In fact, the updating formula in
Equation 8.6 implies the following:
Fact 8.1 Let the outstanding principal be determined by Equation 8.6 and let n be the
number of mortgage payments at time t. Then, the outstanding principal is
n
Lt
=
i= 1
C
(1 +
= C ×A×
(8.7)
rm
12 i
12 )
1 − An
1−A
(8.8)
where A is given in Equation 8.2.4
It is convenient to think about a mortgage as a special bond with only regular coupons,
and no final lump sum principal payment. As with any bond, as interest rates move, the
bond (mortgage) changes in value. In particular, the homeowner may compute the value
of his debt to the bank by using the standard pricing formula
n
Value of mortgage debt = P (t) =
i=1
4 The
C
(1 +
m (t, T )/12)i
r12
i
step from Equation 8.7 to Equation 8.8 is justified by the fact that for A < 1,
− A).
A n )/(1
n
i= 1
A i = A × (1 −
292
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
Figure 8.3 Scheduled Principal Balance, Scheduled Interest, and Principal Payments
5
3
Panel A: Scheduled Principal Balance
x 10
2.5
Dollars
2
1.5
1
0.5
0
0
5
10
15
Years
20
25
30
25
30
Panel B: Scheduled Interest and Principal Payments
2000
Dollars
1500
1000
500
0
Scheduled Interest
Scheduled Principal
Total Coupon
0
5
10
15
Years
20
MORTGAGES AND THE PREPAYMENT OPTION
293
m
where n is the number of payments left at time t and r12
(t, Ti ) are appropriate spot rates,
related to the current term structure of mortgage rates. It is important to note the distinction
between the fixed mortgage rate rm
12 , which is fixed during the life of the mortgage, and the
m
(t, Ti ) which should be used to discount future cash flows.
term structure of spot rates r12
This term structure of interest rates used to discount future cash flows is not the same as,
but it is related to the Treasury curve. Chapters 12 and 13 discuss further the pricing of
mortgages and mortgage backed securities in formal term structure models. In this chapter,
we instead look at simple models to value mortgage backed securities, and their risk.
Consider now a homeowner: At every t the homeowner can compare the amount of
principal left to pay, Lt , with the market value of his liability to the bank P (t). As in any
bond, P (t) increases in value whenever interest rates decline. Thus, when interest rates
decline, the market value of the mortgage owner’s liability P (t) increases to a higher level
than the principal remaining Lt , and it becomes in the homeowner’s interest to refinance the
mortgage. That is, the homeowner can simply extinguish the old liability and open a new
mortgage with P n ew (t) = Lt . This latter mortgage will have a lower monthly payment C,
as the mortgage rate is lower than before.
EXAMPLE 8.1
The years 2001 to 2003 saw record refinancing levels: When the Federal Reserve
acted to prevent a recession by lowering the Federal funds rate from 6% to less than
1%, it set in motion a chain reaction in which banks were able to lower mortgage rates
as well, as their own funding cost decreased substantially. The fixed 30-year mortgage
rate decreased during this period from 8.6% to 5.83%. The one-year adjustable rate
dropped even more, from about 7% in 2000 to 3.76% in 2003. Homeowners had
then the choice between keeping the old higher mortgage rate or refinancing at lower
rates. The lower level of mortgage rates generated a large wave of refinancing, as
shown in Figure 8.4. This figure reports the Mortgage Bankers’ Association (MBA)
refinancing index from early 1990 to late 2007.5 The MBA refinancing index is
based on the number of applications for refinancing, and it is computed from a
weekly survey. The large spike in 2002 - 2003 clarifies the relation between the level
of interest rates (low) and the decision of homeowners to refinance their mortgages.
8.2.1
The Risk in the Prepayment Option
Why is the prepayment such a big issue? The mortgage rate that the bank receives from
the mortgage owner is the return on capital on the bank’s investment. The bank would like
to receive this rate of return for as long as possible after the mortgage is made. However,
if the mortgage owner has the choice of closing the mortgage (prepaying), the bank will
miss this lucrative rate of interest. In particular, because the prepayment will occur mainly
when interest rates decline, the bank that receives the capital back won’t be able to reinvest
it at the same rate of return. In other words, it will lose the mortgage rate. Of course,
5 The
historic time series of the MBA refinancing index is from Bloomberg.
294
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
Figure 8.4
Refinancing and the Federal Funds Rate
10000
10
Refinancing
Index
8000
8
6000
6
4000
4
2000
2
0
1990
1992
1994
1996
1998
2000
2002
2004
2006
Federal Funds Rate (%)
Refinancing Index
Fed Funds Rate
0
2008
Source: Federal Reserve and Bloomberg.
anticipating the prepayment, the bank will increase the mortgage rate to start with, so that
it obtains partial compensation for the risk of prepayment.6
Note that prepayment risk is fundamentally different from default risk. While in the case
of default on a security the bank does not receive its money back, in the case of prepayment
the bank actual does receive its money back, although perhaps at a time in which investment
opportunities are not optimal. Interestingly, homehowners’ defaults actually have an impact
on the timing of prepayments in the case of agency RMBS, because if a homeowner defaults
on his or her mortgage payment, the agency must step in and pay back the mortgage amount
to the RMBS investors, thereby triggering prepayment.
8.2.2 Mortgage Prepayment
The general level of interest rates is an important factor of prepayment. However, it is not
the only one. There are numerous reasons why homeowners pay mortgages early. What
other factors determine prepayment?
1. Seasonality: Summers are characterized by large prepayments, as this is the period
in which people move from one place to another for various reasons. Typically, these
prepayments show up in payments to investors by the end of summer or early fall, as
there are some processing delays.
6 See Chapter 12 for a discussion of the way a mortgage rate should be increased to take into account the prepayment
risk that the bank bears.
MORTGAGE BACKED SECURITIES
295
2. Age of mortgage pool: As shown in Figure 8.3, young mortgages are characterized by
large interest rate payments and low principal. By paying early (whenever possible)
homeowners can save the interest rate payments. Because refinancing is costly,
however, homeowners tend not to refinance new or recently refinanced mortgages
right away, implying a slow prepayment rate in the first few years.
3. Family circumstances: Defaults, disasters, or sale of the house.
4. Housing prices: If the property value of a house declines, it is more difficult to
refinance, and thus prepayments tend to decline. Vice versa, if the property value
of a house increases, the homeowner sometimes takes equity out by refinancing the
loan, inducing a prepayment. The availability of home equity lines, however, reduces
homeowners’ incentives to refinance and thus prepayments.
5. Burnout effect: Mortgage pools heavily refinanced in the past tend to be insensitive
to interest rates. The reason behind the burnout effect is subtle, and it runs as
follows: If a mortgage pool has been heavily refinanced in the past, it must be the
case that most or all of the homeowners that could take advantage of refinancing
opportunities already did so in the past, and thus they are no longer in the pool. The
only homeowners left in such a mortgage pool are those who could not take advantage
from past refinancing possibilities, which makes them less likely to take advantage
of new refinancing opportunities as well, such as a further decline in interest rates.7
The mortgage pool then becomes less sensitive to changes in interest rates when
heavy refinancing activity already occurred in the past.
8.3
MORTGAGE BACKED SECURITIES
Mortgage backed securities derive their characteristics from the features of the mortgages
underlying the pool. Three quantities are particularly important in determining the value
of a mortgage backed security:
1. The Weighted Average Maturity of the mortgages in the pool (WAM).
2. The Weighted Average Coupon of the mortgages in the pool (WAC).
3. The speed of prepayments.
The average maturity and average coupon of a mortgage backed security pool are
relatively simple concepts: For each mortgage in the pool we can compute the time to
maturity and the coupon, and then WAM and WAC are simply the weighted averages of
time and coupon, where the weights are the relative size of each mortgage.
The speed of prepayment is the only concept that is a little elusive. The industry practice
is to refer to some average prepayment rates to describe the speed of prepayment. In fact,
because as we shall see the value of a mortgage backed security depends on the speed of
prepayment, it has been customary to describe the value of a MBS in terms of its speed of
prepayment. We now introduce standard measures of prepayment, which are useful to then
determine the speed of prepayment.
7 There
are numerous reasons why homeowners may not take advantage of refinancing opportunities, ranging
from impaired credit scores and low property value to simple inattention to current market mortgage rates.
296
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
8.3.1 Measures of Prepayment Speed
There are many measures of prepayment speed. The industry practice is to use relatively
simple measures of prepayment mainly to describe, perhaps with a single number, the
expected profile of future cash flows. For instance, if we expect no prepayments for a
while, then we can expect cash flow far in the future. If instead we believe that the current
market is characterized by a high speed of prepayments, then we should expect the MBS
cash flows in the near future and less in the future, as when the mortgage is paid back, no
further coupons will be paid. In this chapter we examine only the most common measures
of prepayment.
8.3.1.1 Constant Maturity Mortality This measure of prepayment is quite basic in
nature: It assumes there is constant probability that the mortgage will be prepaid after the
next coupon. If p is this probability, we have
Pr (Prepayment at time t = 1)
= p
Pr (Prepayment at time t = 2)
=
(1 − p) p
Pr (Prepayment at time t = 3)
=
(1 − p) p
2
The measure p is monthly, as coupon payments are made monthly. Generally, the industry
uses an annualized rate, called the conditional prepayment rate (CPR), obtained from p
as follows:
Pr (Survival up to time t = 12) = (1 − p)
12
= (1 − CP R)
Thus, the CPR is computed from p as
12
CP R = 1 − (1 − p)
and vice versa
1
p = 1 − (1 − CP R) 1 2
This measures the speed at which prepayment occurs.
8.3.1.2 PSA Experience An industry benchmark in the description of prepayment
speed is provided by the 100% PSA. The 100% PSA, established by the Public Securities
Association (PSA), makes the following assumptions:
1. CPR = 0.2% of the principal is paid in the first month;
2. CPR increases by 0.2% in each of the following 30 months; and
3. CPR then levels off at 6% until maturity.
This measure is simple, as it makes the amount of prepayment depend only on the age of
the mortgage pool. It is an industry benchmark, or convention, to express the current belief
about the speed of prepayments. By scaling up or down the CPR in the PSA description,
we obtain faster or slower speeds of prepayments. For instance, Figure 8.5 plots the 100%
PSA line, together with the 150% and 200% PSA. In the next section we provide examples
of the use of PSA for MBS pricing.
MORTGAGE BACKED SECURITIES
Figure 8.5
297
PSA Prepayment Convention
140
Conditional Prepayment Rate CPR (%)
120
100
80
60
40
100 PSA
20
150 PSA
200 PSA
0
8.3.2
0
5
10
15
Age of Mortgage
20
25
30
Pass-Through Securities
A pass-through security is the simplest mortgage backed security: It represents a claim to a
fraction of the total cash flow that is flowing from the homeowners to the pool of mortgages.
This simple structure implies that all investors in pass-through securities are exposed to the
prepayment risk.
EXAMPLE 8.2
Consider a MBS pass through with principal $600 million. The original mortgage
pool has a WAM = 360 months (30 years), and WAC = 6.5%. The pass-through
PT
= 6%, lower that the average coupon rate of the
security pays a coupon equal to r12
mortgage pool, both to ensure there is enough cash available for coupon payments,
and also to provide a compensation for the MBS issuer (e.g., Fannie Mae or Freddie
Mac).
How do we compute the value of the pass through? We can use the PSA level to
determine the speed of prepayment, and therefore the timing and size of future cash
flows. In particular, given a PSA level, for instance 200% PSA, we obtain the CP Rt
for each month t and thus the corresponding monthly prepayment rate pt :
pt = 1 − (1 − CP Rt )1/12
(8.9)
Given that the PSA level determines exactly the amount of principal that is paid
back, we can compute the value of the pass-through security by first computing the
sequence of cash flows, and then, treating these as certain cash flows from a highly
rated company, we discount them to today using the appropriate discount rate. Note
298
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
that Agency MBS are essentially default risk free, implying that the coupons will be
paid to the investors.
To compute the sequence of cash flows, consider a given time t during the life of
the mortgage pool in which Lt is the outstanding principal at the beginning of the
period. From this value, we can compute the following quantities for time t:
Mortgage interest payment:
Scheduled principal:
Principal prepayment:
It
P aytschedu led
P aytpr epaid
m
r12
× Lt
12
= Ct − It
=
= pt × Lt
(8.10)
(8.11)
(8.12)
Given the scheduled principal payments and prepayments, we can finally update both
the outstanding principal and the total coupon flow at the beginning of the following
month t + 1:
Lt+ 1
=
Lt − P aytschedu led − P aytpr epaid (8.13)
Update of scheduled coupon: Ct+ 1
=
(1 − pt ) × Ct
Outstanding principal:
(8.14)
The first Equation 8.13 implies that the new total principal is equal to the previous
month principal minus scheduled and unscheduled principal payments. The prepayment of mortgage also decreases the total flow from coupon, as shown in Equation
8.14. In particular, the new total flow from the pool coupons equals the previous
month coupon flow adjusted for the fraction of prepaid mortgages. For instance, if
100% of homeowners prepay their mortgages at time t, then pt = 1, and the coupon
at time t+1 is zero, as we would expect. Conversely, if nobody repays the mortgages,
then pt = 0, and Ct+ 1 = Ct , that is, the total coupon flow is constant.
These calculations are shown in Table 8.3 for the first 36 months. In Table
8.3, the first column reports the month and the second column shows the constant
prepayment rate (CPR) implied by a 200% PSA. The third column displays the
monthly prepayment rate, as from Equation 8.9. Using these values, Column 4
computes the coupon Ct as in Equation 8.14, except for its first entry (first row),
which instead uses the coupon implied by the weighted average coupon (WAC), the
weighted average maturity (WAM) and the initial principal balance, according to
the formula in Equation 8.3.8 . Column 5 reports the mortgage interest payment,
computed using Equation 8.10. The principal scheduled and the principal prepaid
are computed next, using Equations 8.11 and 8.12, respectively. The total cash flow
of the pass-through security does not depend though on the mortgage interest rate,
but on the pass-through interest rate rP T . Thus, Column 8 reports the pass-through
interest payment, computed as:
Pass-through interest payment: ItP T =
PT
r12
× Lt
12
(8.15)
The pass-through interest rate, scheduled mortgage principal, and mortgage prepayment sum up to form the total cash flow of the pass-through security at the given
month
(8.16)
Total cash flow: CFt = It + P aytschedu led + P aytpr epaid
8 More
specifically, C = L/(
W AM
i= 1
A i ) where A = 1/(1 + W AC/12).
MORTGAGE BACKED SECURITIES
Table 8.3
Month
i
(1)
CPR
p
(2)
(3)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
0.20%
0.40%
0.60%
0.80%
1.00%
1.20%
1.40%
1.60%
1.80%
2.00%
2.20%
2.40%
2.60%
2.80%
3.00%
3.20%
3.40%
3.60%
3.80%
4.00%
4.20%
4.40%
4.60%
4.80%
5.00%
5.20%
5.40%
5.60%
5.80%
6.00%
6.00%
6.00%
6.00%
6.00%
6.00%
6.00%
0.03%
0.07%
0.10%
0.13%
0.17%
0.20%
0.24%
0.27%
0.31%
0.34%
0.37%
0.41%
0.44%
0.48%
0.51%
0.55%
0.59%
0.62%
0.66%
0.69%
0.73%
0.76%
0.80%
0.84%
0.87%
0.91%
0.95%
0.98%
1.02%
1.06%
1.06%
1.06%
1.06%
1.06%
1.06%
1.06%
299
Computations: Cash Flows of Pass-Through Security
Coupon Mortgage Principal
Interest Scheduled
(4)
(5)
(6)
3.79
3.79
3.79
3.78
3.78
3.77
3.77
3.76
3.75
3.74
3.72
3.71
3.69
3.68
3.66
3.64
3.62
3.60
3.58
3.55
3.53
3.50
3.48
3.45
3.42
3.39
3.36
3.33
3.29
3.26
3.23
3.19
3.16
3.12
3.09
3.06
3.25
3.25
3.24
3.23
3.23
3.22
3.21
3.20
3.19
3.17
3.16
3.15
3.13
3.11
3.09
3.08
3.06
3.03
3.01
2.99
2.97
2.94
2.92
2.89
2.86
2.83
2.81
2.78
2.75
2.71
2.68
2.65
2.62
2.59
2.56
2.53
0.54
0.55
0.55
0.55
0.55
0.55
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.56
0.55
0.55
0.55
0.55
0.54
0.54
0.54
0.53
0.53
0.53
Principal
Prepaid
(7)
Pass-Through
Interest
(8)
Total
Cash Flow
(9)
Principal
Lt
(10)
Discount
Z (0, T )
(11)
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
1.79
1.99
2.18
2.38
2.57
2.75
2.94
3.12
3.30
3.48
3.65
3.82
3.99
4.15
4.31
4.47
4.62
4.77
4.91
5.05
5.18
5.31
5.25
5.19
5.13
5.07
5.01
4.95
3.00
3.00
2.99
2.99
2.98
2.97
2.96
2.95
2.94
2.93
2.92
2.90
2.89
2.87
2.86
2.84
2.82
2.80
2.78
2.76
2.74
2.72
2.69
2.67
2.64
2.62
2.59
2.56
2.53
2.51
2.48
2.45
2.42
2.39
2.36
2.34
3.74
3.94
4.14
4.34
4.53
4.73
4.92
5.11
5.30
5.48
5.66
5.84
6.02
6.19
6.36
6.53
6.69
6.84
7.00
7.15
7.29
7.43
7.56
7.69
7.82
7.94
8.05
8.16
8.27
8.36
8.27
8.18
8.08
7.99
7.90
7.81
599.26
598.31
597.16
595.81
594.25
592.50
590.54
588.39
586.03
583.48
580.73
577.80
574.67
571.35
567.85
564.16
560.29
556.25
552.04
547.65
543.10
538.38
533.51
528.48
523.31
517.98
512.52
506.92
501.19
495.33
489.54
483.81
478.15
472.55
467.01
461.53
0.9979
0.9958
0.9938
0.9917
0.9896
0.9876
0.9855
0.9835
0.9814
0.9794
0.9773
0.9753
0.9733
0.9713
0.9692
0.9672
0.9652
0.9632
0.9612
0.9592
0.9572
0.9552
0.9532
0.9512
0.9492
0.9473
0.9453
0.9433
0.9414
0.9394
0.9375
0.9355
0.9336
0.9316
0.9297
0.9277
300
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
The total cash flow is provided in Column 9 of Table 8.3. Finally, the last column
updates the new amount of principal remaining, according to Equation 8.13.
The value of the pass-through security is then obtained by treating the cash flows in
Column 9 as known amounts of dollars in the future, without any uncertainty. Thus,
they are simply discounted by using a discount function appropriate for this credit
rating. Some of these pass-through securities are default free: For instance, Ginnie
Mae originated pass-through securities are guaranteed against default by the full
faith of the U.S. Government, and thus investors in these securities are not subject to
default. Similarly, over the years market participants believed also that the securities
issued by Fannie Mae and Freddie Mac had small default risk, as it was believed that
the U.S. governmnet would bail these agencies out in case of financial troubles.9 If
we think of these securities having zero or small default risk, then we can use the
Treasury discount curve to discount these cash flows. Assume for instance a flat term
structure with constant (continuously compounded) 5% yield. The corresponding
discount Z(0, T ) is reported in the last column. For this exercise, the value of the
pass-through security is $635 million, above its par value of $600 million.
8.3.3
The Effective Duration of Pass-Through Securities
Pass-through MBS have some peculiarities in terms of their sensitivity to interest rate
movements. The reason is that as interest rates decline, homeowners may decide to
refinance their mortgages. This refinancing activity increases the conditional prepayment
rate. That is, it increases the PSA, which could easily move from 200%, for instance, to
300%. This change has an impact on the sensitivity of the pass-through security to interest
rates. The next example illustrates the issue.
EXAMPLE 8.3
Consider the pass-through MBS discussed in Example 8.2. In that example, we
assume that the current PSA level is PSA = 200%. Consider now the calculation of
its duration assuming first that the PSA level is unaffected by the change in interest
rate. In this case, because the pass-through MBS has a constant coupon, we can
compute its duration as in Chapter 3, Section 3.2.3, and obtain a duration value of
D = 5.83.
Consider now the case in which if the interest rate moves down from 5% to 4.50%,
the PSA increases from 200% to 250%, while if the interest rate moves up from 5%
to 5.50%, the PSA decreases from 200% to 150%. How can we compute the duration
of the pass-through security in this case? We can apply the definition of duration in
Definition 3.2 in Chapter 3, namely
D=−
1 dP
P dr
9 This belief proved correct ex post, as both Fannie Mae and Freddie Mac went into conservatorship in September,
2008 as a consequence of the mortgage backed security market turmoil of 2007 and 2008.
MORTGAGE BACKED SECURITIES
301
Given this definition, we can approximate the duration of the pass-through security
while taking into account the impact of interest rate changes on PSA levels as follows:
D≈−
1 P (+50bps) − P (−50bps)
P
2 × 50bps
(8.17)
where P = $634.76 is the current value of the pass-through security, obtained in
Example 8.2, and P (+50bps) and P (−50bps) are the values of the same passthrough security when we increase and decrease the interest rate by 50 basis points,
respectively, and the PSA levels accordingly. By carrying out exactly the same
computations as in Example 8.2 but for the two cases in which the interest rate is
either 5.5% or 4.5%, and the PSA level is 150% or 250%, respectively, we find
P (+50bps) = $619.13
P (−50bps) = $647.45
Substituting these values into Equation 8.17 we obtain
D≈−
$619.13 − $647.45
1
= 4.46
$634.76
2 × 50bps
This duration is much smaller than the duration that we obtained when we neglected
the impact on the change in PSA due to changes in interest rates, which was 5.83.
Missing the impact of interest rate variation on the speed of prepayment may grossly
overstate the sensitivity of the pass-through security to changes in interest rates, and
thus the performance of any duration-based hedging activity.
This example highlights a potential pitfall in the risk assessment of mortgage backed
securities, namely, the necessity to take into account the variation in prepayment speed due
to the change in interest rates. Because of this variation in prepayment speed, analytical
formulas for the computation of duration are not available, and thus we must rely on
approximations, as illustrated in Example 8.3. This approximation is called effective
duration:
Definition 8.1 The effective duration of a mortgage backed security is given by the
formula
1 P (+x bps) − P (−x bps)
(8.18)
D≈−
P
2 × x bps
where P is the current price of the MBS, and P (+x bps) and P (−x bps) are the prices
of the same security after we shift upward or downward the yield curve by x basis points,
respectively. In this computation, the price of the MBS takes into account the variation in
the prepayment speed induced by the variation in interest rates.
To compute the effective duration of a MBS we need the predicted variation of the pool
prepayment speed that is induced by a parallel shift in the term structure of interest rates.
Market participants use sophisticated models to predict such variation. The case study in
Section 8.7 futher illustrates the computation of the effective duration of a pass-through
security by relying on market participants’ forecasts of the change in prepayment speed.
In addition, the case study also illustrates the use of historical market data on MBS prices
and interest rates to estimate the effective duration of a pass-through security.
302
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
8.3.4 The Negative Effective Convexity of Pass-Through Securities
A second feature of prepayments is to induce a negative convexity of the MBS price with
respect to interest rates. Typically, fixed income securities increase substantially in price as
interest rates decline (see Chapter 4). However, if the interest rate decline makes the rate of
prepayment increase, then the price of the security does not increase as much, generating
a negative convexity.
EXAMPLE 8.4
Consider Table 8.4. The first two columns show the impact of interest rates on the
price of the pass through in Example 8.2, when the speed of prepayment is kept
constant at 200% PSA. The price of the pass through increases substantially as rates
decline, as we know is typical of any interest rate security. Suppose now that as
the interest rate declines, the speed of prepayments increases. As an example, the
second three columns of Table 8.4 show the case in which as the interest rate declines
from 6% to 2% (Column 4) the PSA increases from 100% to 500%. Similarly, if the
interest rate increases, the PSA declines. The higher prepayment rate moves the price
of the pass through closer to its principal value, which is $600 million. Indeed, if
everybody prepayed at the same time, the value of the pass through would be exactly
equal to $600 million, as it would equal its principal amount. In reality, even with
dramatic decreases in interest rates, many households do not prepay their mortgages.
Still, even if the PSA increases only to 500%, the value of the pass-through security
goes up only to $688 million, compared to $765 million in the case of constant PSA.
Figure 8.6 plots the value of the pass-through security in Table 8.4 with respect to
the interest rate for the two cases in which the PSA is constant at 200% (the dotted
line) and the case in which the PSA increases as interest rates decline (the solid line).
As is clear from the picture, the slope of the solid line is smaller when interest rates
are low than when they are high. This implies that the pass-through security with
changes in PSA is less sensitive to interest rate variation when interest rates are low,
while it is more sensitive when interest rates are high, as shown in Table 8.4. In other
words, the pass-through security in Figure 8.6 displays negative convexity.
The negative convexity of pass-through securities illustrated in the previous example, and
further documented in Section 8.3.5 below, is a source of risk for investors. Mechanically,
the source of the negative convexity is the higher prepayment that is induced by lower
interest rates, which pushes the pass-through price toward its outstanding principal amount.
Economically, the investor in a pass-through security has implicitly written American call
options to the homeowners of the mortgages in the pool (see Chapter 6): When interest
rates decline and thus the value of the loan increases above the outstanding principal,
homeowners exercise the option and repay their mortgages to investors (by refinancing
them). As we know from Figure 6.2 in Chapter 6, the payoff profile of a short call option
is negatively convex. Thus, adding these short call options to the long position in the pool
of mortgages generates the negative covexity of the pass-through security.
Indeed, pass-through securities have a higher yield than equivalent Treasury securities,
even if they have no default risk when issued by Ginnie Mae, for instance. We can see
this from Table 8.5. This table reports a sample of Ginnie Mae (GNMA) pass-through
prices at selected dates at which they were trading close to par-value (100). Because the
MORTGAGE BACKED SECURITIES
303
Table 8.4 Pass-Through Security Value versus Interest Rate and PSA
Constant PSA = 200%
Increasing PSA with Lower Interest Rate
Interest Rate
Value
Interest Rate
PSA
Value
2.00%
2.50%
3.00%
3.50%
4.00%
4.50%
5.00%
5.50%
6.00%
6.50%
7.00%
7.50%
8.00%
8.50%
9.00%
764.57
740.00
716.72
694.63
673.66
653.73
634.76
616.71
599.50
583.08
567.41
552.44
538.12
524.42
511.30
2.00%
2.50%
3.00%
3.50%
4.00%
4.50%
5.00%
5.50%
6.00%
6.50%
7.00%
7.50%
8.00%
8.50%
9.00%
500
450
400
350
300
250
200
150
100
90
80
70
60
50
40
687.80
681.64
674.76
666.97
658.00
647.45
634.76
619.13
599.33
576.96
554.53
532.06
509.54
487.01
464.45
Figure 8.6 The Value of a Pass-Through MBS with Respect to the Interest Rate
800
Constant PSA = 200
Higher PSA with Lower Interest Rate
750
Value of Pass Through
700
650
600
550
500
450
2
3
4
5
6
Interest Rate (%)
7
8
9
304
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
Table 8.5
A Sample of Par Value Ginnie Mae Pass-Through Prices and Treasury Yields
Ginnie Mae Pass-Through
Treasury Constant Maturity Rates
Date
Bid
Ask
Coupon
WAC
WAM
10-Year
20-Year
30-Year
09/26/1997
06/12/1998
08/07/1998
05/05/2006
07/28/2006
08/17/2007
100.0313
100.0313
100.0000
100.1250
99.96875
100.0938
100.0000
100.0000
99.96875
100.0938
99.9375
100.0625
7.0
6.5
6.5
6.0
6.0
6.0
7.5
7.0
7.0
6.5
6.5
6.5
315
324
327
315
318
320
6.08
5.43
5.40
5.12
5.00
4.68
6.43
5.75
5.72
5.35
5.17
5.06
6.37
5.66
5.63
5.20
5.07
5.00
Source: Bloomberg and Federal Reserve Board.
pass-through security is trading around par value, we can compare directly its coupon rate,
in Column 4, to the yields of similar “close to par value” Treasury securities, given by
the Treasury constant maturity rates in Columns 7 to 9.10 . Column 6 of the table reports
the weighted average maturity of the underlying pool, providing an idea of the maximum
maturity of the pass-through security. Comparing the coupon rates of the pass-through
securities in Column 4 to the constant maturity rates in Columns 7 to 9, it is clear that
the pass-through securities have a higher coupon rate, even if they trade at the same price
(par value). We can interpret this spread over Treasuries as the option premium implicitly
received by pass-through MBS investors from the sale of call options.11
Why is negative convexity such a big issue? To answer this question, we should
recall the discussion in Section 4.1.4 of Chapter 4. In that section, we argued that the
positive convexity of bonds is good news for bond investors. The reason, recall, was that
because interest rates keep moving day-by-day in an unpredictable manner, the positive
convexity of a bond generates positive capital gain returns, on average. Negative convexity
generates instead negative capital gain returns, on average, as interest rates move randomly
through time. Because of this average negative capital gain return, investors in passthrough securities require a higher coupon rate on the pass-through securities compared to
equivalent Treasuries. Thus, an equivalent way to interpret the higher yield of pass-through
securities with respect to Treasuries is as compensation for the average capital loss due to
negative convexity.
How can we measure the negative convexity of pass-through securities? Because of the
changes in prepayment speed there are no analytical formulas available. Therefore we must
rely on an approximation, similar to the effective duration in Definition 8.1. In particular,
10 According
to the U.S. Treasury Web site, the U.S. Treasury computes the constant maturity rates by interpolating yields only of recently issued (on-the-run) Treasury securities, as they tend to trade around par value.
See http://www.ustreas.gov/offices/domestic-finance/debt-management/interest-rate/yieldmethod.html (accessed
on April 29, 2009).
11 We should mention that part of the spread may be due to other reasons, such as the difference in liquidity
between GNMA pools and Treasury securities.
MORTGAGE BACKED SECURITIES
305
recalling the definition of convexity in Definition 4.2 in Chapter 4, namely
C=
1 d2 P
P dr2
we obtain the approximating formula outlined in the following definition.
Definition 8.2 The effective convexity of a mortgage backed security is given by the
formula
1 P (+x bps) + P (−x bps) − 2 × P
C≈
(8.19)
P
(x bps)2
where P is the current price of the MBS, and P (+x bps) and P (−x bps) are the prices
of the same security after we shift upward or downward the yield curve by x basis points,
respectively. In this computation, the price of the MBS takes into account the variation in
the prepayment speed induced by the variation in interest rates.
As an illustration of the formula in Equation 8.19, consider again Example 8.4.
EXAMPLE 8.5
Consider again Example 8.4. We want to compute the convexity of the pass-through
security when the current interest rate is r = 5%. Referring to Table 8.4, we can
apply the formula in Equation 8.19 with x = 50. In this case, we have
C
≈
=
=
1 P (+50 bps) + P (−50 bps) − 2 × P
P
(50 bps)2
1 619.13 + 647.45 − 2 × 634.76
634.76
(50 bps)2
−184.89
As expected from Figure 8.6 and the previous discussion, the effective convexity of
the MBS is negative.
We further illustrate the characteristics of pass-through securities and look at the data in
the next section.
8.3.5
The TBA Market
The secondary market for pass-through securities is quite active. In particular, Ginnie Mae,
Fannie Mae, and Freddie Mac regularly issue new pass-through securities, whose prices
become the market reference. Most of the MBS trading is on a “To-Be-Announced” (TBA)
basis, which means that traders do not know the exact composition of the pool underlying
the trade when they bid for the security. These pools are quite homogenous, however, and
differences are small. Essentially, the TBA market can be thought of as a forward market,
in which two counterparties agree today that they will exchange in the future cash for a
pass-through pool. However, at the time of the trade the mortgage pool has not been set
up by the originators. Yet, since the pool has been sold already for a given price, banks are
able to offer lock-in mortgage rates to homeowners, as they know they would be able to
306
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
later sell the mortgage to the given pool in the secondary market.12 Liquidity in the TBA
market has then important benefits for the borrowers themselves, as they would receive a
better mortgage rate if the bank knows that it will not have to bear the risk of default of the
new borrower. Only Ginnie Mae, Fannie Mae, and Freddie Mac pass-through securities
are allowed to trade in the TBA market.
Panel A of Figure 8.7 plots quotes on GNMA 7 from January, 1995 to November, 2008.
The GNMA 7 prices are for a generic pass through issued by Ginnie Mae with a 7% coupon.
Note that these quotes are for the latest issued GNMA with 7% coupon, and so the prices
reported in the figure are not for the same pass through over time. In particular, the maturity
is not decreasing over time, as it should if the underlying pool of mortgages remained the
same. Still, because of the TBA market and the relatively uniformity of underlying pools,
trades occur at these prices even if the underlying security changes slightly from one time to
the other. Panel A also reports the 30-year mortgage rate, which moves opposite to the price
of GNMA pass through. The interesting pattern to notice is that while in the first part of the
sample, when the average mortgage rate was around 7.5%, large variations in the mortgage
rate resulted in large variation in GNMA prices, this behavior is less evident in the second
part of the sample, when mortgage rates averaged less than 6%. That is, the two lines still
move in opposite directions, but the increase in prices due to a decrease in the mortgage
rate is smaller when the mortgage rate is low. This behavior is due to the negative convexity
of GNMA pass throughs induced by the prepayment option, as discussed earlier in Section
8.3.4. That is, as mortgage rates decrease, homeowners refinance their mortgages and thus
prepay the old mortgages. The prepayment makes the mortgage pool value get closer to
the principal value, generating a negative convexity.
The negative convexity of GNMA price with respect to the mortgage rate is shown
in Panel B of Figure 8.7, which reports the scatterplot of the GNMA prices versus the
mortgage rates. More specifically, each star (“+”) in the scatterplot represents a realized
combination of GNMA price and mortgage rate, as they appear in Panel A of the same
figure. For example, the data point in the far right of the plot has a price / rate combination
of (89.5625, 0.0922), which is the first observation in the data sample, appearing at the
beginning of the time series plot in Panel A (i.e., January 1995). As Panel B shows, when
the mortgage rate declines, the price of GNMA rises, but at an increasingly lower rate.
Indeed, the figure also displays a solid line that is fitted through the data to best characterize
the relation between prices and mortgage rates. This solid line clearly shows the negative
convex relation between GNMA prices and mortgage rates in the data.13 Figure 8.8 plots
the PSA measure of the GNMA 7 pool and the mortgage rate. As can be seen, a lower
mortgage rate corresponds in average to a higher prepayment speed, which in turn explains
the negative convexity of MBS prices.
8.4 COLLATERALIZED MORTGAGE OBLIGATIONS
Collateralized mortgage obligations (CMOs) are securities with structures that are more
complex than pass through securities.14 The main idea behind this type of security is that
12 See
e.g., “General Description of the TBA Market,” by the Security Industry and Financial Market Association (SIFMA), available on the Web site http://www.sifma.org/capital markets/TBA-MBS.shtml, or “Enhancing
Disclosure in the Mortgage Backed Securities Market,” January 2003, SEC Staff Report.
13 More specifically, we compute the solid line in Panel B of Figure 8.7 through a polynomial regression. We first
4
regress P t = α 0 +
β (r m )i + t and then plot the fitted value of the regression.
i= 1 i t
14 CMOs are also called REMICs for Real Estate Mortgage Investment Conduit.
307
COLLATERALIZED MORTGAGE OBLIGATIONS
Figure 8.7 GNMA 7 Prices and Mortgage Rates
Panel A: Time Series
110
0.1
0.09
GNMA
Price
100
0.08
95
0.07
Mortgage
Rate
90
85
1994
1996
1998
2000
0.06
2002
2004
0.05
2008
2006
Panel B: Convexity
110
Data
Fitted Curve
GNMA Price
105
100
95
90
85
0.05
0.055
0.06
0.065
0.07
0.075
Mortgage Rate
Data source: Bloomberg and Federal Reserve Board.
0.08
0.085
0.09
0.095
Mortgage Rate
GNMA Price
105
308
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
Figure 8.8 GNMA 7 PSA and Mortgage Rates
Panel A: Time Series
0.1
GNMA PSA
Mortgage
Rate
GNMA
PSA
1000
0.08
500
0
1994
0.06
1996
1998
2000
2002
2004
0.04
2008
2006
Panel B: Scatterplot
1200
Data
Fitted Curve
GNMA PSA
1000
800
600
400
200
0
0.05
0.055
0.06
0.065
0.07
0.075
Mortgage Rate
Data source: Bloomberg and Federal Reserve Board.
0.08
0.085
0.09
0.095
Mortgage Rate
1500
COLLATERALIZED MORTGAGE OBLIGATIONS
309
it offers different levels of exposure to prepayment risk. This in turn affects its risk and
return characteristics, including its duration and sensitivity to interest rate variation. The
advantage of structuring CMOs with different risk return characteristics is to appeal to
different investors’ clienteles, which may increase the liquidity of these securities. Indeed,
CMOs must satisfy a number of requirements in order to also receive a high credit rating,
a property that makes them suitable for investment from a larger set of potential investors
and thus increases liquidity. While agency CMOs are perceived default free, non-agency
CMOs must instead be over-collateralized to obtain a high credit rating, meaning that the
total value of securities issued is much lower than the value of the assets in the pool. In
addition, many CMOs pay quarterly coupons, instead of monthly ones, making them more
in line with other investment vehicles.
In the remainder of this chapter we go over the most common types of structures.
8.4.1
CMO Sequential Structure
The first step in a sequential structure CMO is to divide the total principal into smaller
groups, which are called “Tranche A,” “Tranche B,” “Tranche C” and so on. These tranches
then receive the following cash flow: (i) a fixed coupon rate payment, in percentage of
the tranche principal; (ii) sequentially, all of the principal payments, scheduled or prepaid,
up to the point at which the whole principal of the tranche is paid out. At that point,
the investor buying that particular tranche has received all of his or her money back, and
thus the tranche is fully retired. The principal payments, scheduled and unscheduled, then
start flowing to the next tranche in the sequence (from which we get the name sequential
structure).
Sometimes, a “Tranche Z” is also added. This tranche receives no cash flows (the letter
“Z” stands for zero, as zero coupon bond), but the coupon is accrued over time to its
principal. Once all the other classes have been retired, then principal payments go to the
class Z. The tranche Z allows the issuers to lessen the impact of prepayments on the early
tranches by shifting cash flows between tranches, with the Z getting whatever is left over
when the others are paid off. The following example illustrates the methodology.
EXAMPLE 8.6
Let’s consider Example 8.2 again. This time, however, we subdivide all of the cash
flows generated by the pass-through security (Column 9 in Table 8.3) into smaller
cash flows into four classes, A, B, C, and D. For instance, suppose that Tranche A
has $250 million principal, Tranche B $150 million, Tranche C $125 million and
Tranche D $75 million. Let each tranche have a coupon of 6%, as the original pass
through does. Table 8.6 illustrates the procedure. The first column is the month, the
second column is the total cash flow of the pass-through security, as seen in Column
9 of Table 8.3. This cash flow is now divided across the four tranches. Initially,
tranche A receives both principal payments and interest. The interest component is
computed as the coupon rate (6%) times the remaining balance in Column 3. The
principal payment is given by the total of scheduled and prepaid principal from the
original pool. For instance, the first principal payment is $0.74 = $0.54 + $0.20,
where $0.54 is the scheduled principal, and $0.20 is the prepaid principal, as shown
in Columns 6 and 7 in Table 8.3. Similarly, in month 2, Tranche A investors receive
310
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
a total of principal equal to $0.95 = $0.55 + $0.40, where $0.55 and $0.40 are the
scheduled and prepaid principal. And so on.
Tranche B to D in these first months are paid the interest only, but no principal
at all. For instance, Tranche B receives an interest of $0.75 = 6%
12 × 150, which
determines all of its cash flow. In the illustration in Table 8.6, Tranche B begins
receiving principal payments in month 60, when all of the principal of Tranche A
($250 million) has been paid back to investors. The procedure to define principal
payments to Tranche B is the same as the one we discussed for Tranche A. For
instance, in month 61 Tranche B receives $4.14 of principal, of which $0.46 is a
scheduled payment and $3.68 is a prepayment. In this illustration, Tranche B is
paid back in month 105, and thus principal payments are made to Tranche C. This
latter tranche is paid back in month 180, and principal payments begin to be made to
Tranche D.
Once the cash flows are determined, we can compute the value of each tranche
by using the standard present value formula. In this case, we obtain that Tranche A
is valued at $256.42 million, Tranche B at $158.42 million, Tranche C at $135.71
million, and Tranche D at $84.21 million. Of course, because the tranche cash flows
sum to the original pass- through cash flows, the sum of the values of tranches is the
same as the value of the pass-through MBS, that is, $634.76 million.
It is important to note that the different allocation of principal payments drastically
changes the sensitivity of the MBS to interest rate movements. For instance, the
effective duration of Tranche A is 1.96, of Tranche B is 3.99, of Tranche C is 6.27
and of Tranche D is 10.04. Since the pass-through security is a portfolio of tranches,
the duration of the pass through equals the weighted average of durations (see Chapter
3), that is, 4.46, as discussed earlier.
The speed of prepayments affects the time at which the various tranches are
paid back. The illustration in Table 8.6 assumes 200% PSA. If for instance we
considered 100% PSA, then Tranche A is retired in month 96 (instead of 60),
Tranche B in month 171 (instead of 105), Tranche C in month 267 (instead of
180). The duration of individual tranches changes accordingly, in this case, up
to 3.70 for Tranche A, 8.15 for Tranche B, 11.39 for Tranche C, and 13.96 for
Tranche D.
8.4.2
CMO Planned Amortization Class (PAC)
The PAC securities are also in tranches: A, B, C and Companion. Tranches A, B, C and so
on receive prepayments according to a prespecified schedule, which is related to PSA levels
chosen ex ante by the MBS issuer. This benchmark PSA-related principal payments is kept
fixed for the duration of the contract. However, since this predefined PSA schedule may
imply prepayments that are higher or lower than the actual prepayments from homeowners,
the difference is absorbed by a Companion Tranche. The implication is that Tranches A,
B, C etc. have deterministic future cash flows and thus they can be priced as any coupon
bond. In contrast, the Companion Tranche absorbs all of the prepayment risk, and thus its
price is highly affected by prepayments.
To make this methodology operational, an upper and lower PSA level is defined first.
Given the principal amount of the tranche, these two PSA levels determine two profiles of
COLLATERALIZED MORTGAGE OBLIGATIONS
311
Table 8.6 Collateralized Mortgage Obligation – Sequential Structure
Total
Tranche A
CF Balance Pri. Int.
CF
Balance
Tranche B
Pri. Int.
CF
Balance
Tranche C
Pri. Int.
CF
1
2
3
4
5
6
7
8
9
10
11
12
3.74
3.94
4.14
4.34
4.53
4.73
4.92
5.11
5.30
5.48
5.66
5.84
250.00
249.26
248.31
247.16
245.81
244.25
242.50
240.54
238.39
236.03
233.48
230.73
0.74
0.95
1.15
1.35
1.55
1.76
1.96
2.16
2.35
2.55
2.75
2.94
1.25
1.25
1.24
1.24
1.23
1.22
1.21
1.20
1.19
1.18
1.17
1.15
1.99
2.19
2.39
2.59
2.78
2.98
3.17
3.36
3.55
3.73
3.91
4.09
150.00
150.00
150.00
150.00
150.00
150.00
150.00
150.00
150.00
150.00
150.00
150.00
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
0.75
125.00
125.00
125.00
125.00
125.00
125.00
125.00
125.00
125.00
125.00
125.00
125.00
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
55
56
57
58
59
60
61
62
63
64
65
66
67
6.29
6.22
6.15
6.08
6.01
5.94
5.87
5.81
5.74
5.68
5.61
5.55
5.48
22.72
18.29
13.91
9.58
5.30
1.06
4.43
4.38
4.33
4.28
4.23
1.06
0.11
0.09
0.07
0.05
0.03
0.01
4.54
4.47
4.40
4.33
4.26
1.07
150.00
150.00
150.00
150.00
150.00
150.00
146.87
142.73
138.64
134.59
130.59
126.63
122.72
3.13
4.14
4.09
4.05
4.00
3.96
3.91
3.87
0.75
0.75
0.75
0.75
0.75
0.75
0.73
0.71
0.69
0.67
0.65
0.63
0.61
0.75
0.75
0.75
0.75
0.75
3.88
4.87
4.81
4.74
4.68
4.61
4.55
4.48
125.00
125.00
125.00
125.00
125.00
125.00
125.00
125.00
125.00
125.00
125.00
125.00
125.00
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
0.63
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
100
101
102
103
104
105
106
107
108
109
110
111
112
3.74
3.70
3.65
3.61
3.57
3.53
3.49
3.45
3.41
3.37
3.33
3.29
3.25
15.51
12.85
10.22
7.61
5.04
2.50
2.66
2.63
2.60
2.57
2.54
2.50
0.08
0.06
0.05
0.04
0.03
0.01
2.74
2.70
2.65
2.61
2.57
2.51
125.00
125.00
125.00
125.00
125.00
125.00
124.98
122.49
120.03
117.60
115.20
112.82
110.48
0.02
2.49
2.46
2.43
2.40
2.38
2.35
2.32
0.63
0.63
0.63
0.63
0.63
0.63
0.62
0.61
0.60
0.59
0.58
0.56
0.55
0.63
0.63
0.63
0.63
0.63
0.65
3.11
3.07
3.03
2.99
2.95
2.91
2.87
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
75.00
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
0.38
175
176
177
178
179
180
181
182
183
184
185
186
187
1.52
1.50
1.48
1.46
1.45
1.43
1.41
1.39
1.38
1.36
1.34
1.33
1.31
5.99
4.87
3.77
2.68
1.61
0.54
1.12
1.10
1.09
1.08
1.06
0.54
0.03
0.02
0.02
0.01
0.01
0.00
1.15
1.13
1.11
1.09
1.07
0.55
75.00
75.00
75.00
75.00
75.00
75.00
74.49
73.45
72.43
71.41
70.41
69.42
68.44
0.38
0.38
0.38
0.38
0.38
0.38
0.37
0.37
0.36
0.36
0.35
0.35
0.34
0.38
0.38
0.38
0.38
0.38
0.88
1.41
1.39
1.38
1.36
1.34
1.33
1.31
Month
Tranche D
Balance Pri. Int.
0.51
1.04
1.03
1.01
1.00
0.99
0.98
0.97
CF
312
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
total principal (scheduled and prepaid) for the tranche, which are determined by using the
same methodology as in Equations 8.10 through 8.16. Denote these total principals
P aythi,total
= P aythi,schedu led + P aythi,pr epaid
P aytlo,total
= P aytlo,schedu led + P aytlo,pr epaid
as they correspond to the high PSA and low PSA. Unlike what we may think at first, it is
not the case that P aythi,total > P aytlo,total all the time. In fact, initially, P aythi,total is
higher than P aytlo,total , as the higher assumed prepayment rate implies a higher cash flow.
However, the higher initial cash flows imply that the principal is depleted more quickly,
and thus, sooner or later, P aythi,total < P aytlo,total . Because of this behavior of planned
cash flows for a given PSA level, we must be careful to determine the promised cash flow to
Tranches A, B, and so forth, so that with high probability, the total amount of prepayments
from homeowners is sufficient to pay for the scheduled PSA-related prepayment. PAC cash
flows are determined by the formula
(8.20)
Promised CF Ctpac = It + min P aythi,total , P aytlo,total
This choice ensures that as long as homeowners’ prepayment speed is within the prespecified
hi and lo PSA levels, there will be enough cash flows coming from homeowners to pay for
the promised cash flow of the tranche. The companion tranche then absorbs any difference
between the total cash flows from the pass through and the one promised to the PAC Tranche
investors. That is
Companion Tranche CF CtC om = CtP T − Ctpac
(8.21)
where CtP T denotes the total cash flow in the original pass-through security. The PAC
profile of total principal payments (scheduled and unscheduled) is then given by
P aytpac = min P aythi,total , P aytlo,total
Since these amounts provide a firm schedule, the amount of PAC principal must equal the
sum of P aytpac from 0 to maturity. That is, we set
360
Lpac
=
0
P aytpac
(8.22)
t= 1
om
= L0 − Lpac
The Companion Tranche principal is set as a residual, so that LC
0
0 . Let’s
look at an example to see this methodology in practice.
EXAMPLE 8.7
Consider Example 8.2 again. Once more, we divide all of the cash flows generated
by this pass through (Column 9 in Table 8.3) into two smaller cash flows, going
to Tranche A and its Companion Tranche. Let’s assume that the PSA range is
PSAhi = 300% and PSAlo = 80%. The calculations above imply that the PAC
Tranche has principal equal to $356.69 million, and the Companion Tranche has
principal equal to $243.31 million.
COLLATERALIZED MORTGAGE OBLIGATIONS
313
Figure 8.9 shows the two cash flow profiles, with the high PSA spiking up initially,
and then declining below to the cash flow profile of the low PSA. The PAC scheduled
cash flow, then, is computed as the minimum between these two cash flows, to make
sure that changes in the speed of prepayments of the original pool do not generate
too small of a cash flow or too large of a cash flow that cannot be absorbed by the
Companion Tranche.
How does the PAC scheduled cash flow depend then on a realized (true) PSA?
That is, the scheduled cash flow is determined by using the issuer’s best guess of
what future real prepayments will be. However, in reality, prepayments depend on
market conditions, interest rates, housing prices and so on. These other factors will
have an impact on the speed of prepayment. Figure 8.10 shows four panels with
the cash flow to the PAC Tranche and the Companion Tranche, under four different
assumptions about the realized (true) PSA. For instance, suppose that the true PSA
is 100%. Panel A of Figure 8.10 shows the profile of the PAC Tranche cash flow,
which equals in fact the one in Figure 8.9. The Companion Tranche has to absorb
any difference between the true cash flow from the pass through (the solid line) and
the one promised to the PAC Tranche holders (the dashed line). The resulting cash
flow of the Companion Tranche is shown as the dotted line.
Consider now the case in which prepayments accelerate, perhaps because the
Federal Reserve decreased the Fed funds rate and mortgage rates declined. Panel B
shows the cash flow profile when the PSA goes up to 200%. This change makes the
total cash flow from the pass through (the solid line) increase substantially. However,
this increase in prepayments does not affect the PAC Tranche cash flow, as its profile
(the dashed line) does not change at all. The impact is instead on the Companion
Tranche, whose cash flow increases substantially. Panel C displays the same plot
but under a realized 300% PSA. In this case for a while the Companion Tranche can
absorb the difference in prepayment between the promised cash flows and the true
cash flows. However, as the Companion Tranche receives principal prepayments, its
principal declines and at some point it is fully retired. This is for the same reason
that tranches A, B and so on are retired in the sequential structure discussed earlier.
As soon as the capital of the Companion Tranche is depleted, the cash flow of the
PAC Tranche equals the one of the pass through. Effectively, with the Companion
Tranche retired, the PAC Tranche reverts back to the original pass through, and thus
prepayment risk is back fully in force.
What happens if the realized (true) PSA is actually above the highest forecasted
PSA? As shown in Panel D of Figure 8.10, in this case, the Companion Tranche’s
principal is repaid very quickly, and thus the cash flow of the PAC Tranche has to jump
suddenly to the one of the pass through. Once again, in this case, the PAC Tranche
cash flows are no longer determined by the original promised schedule, rather they
revert back to the ones of a standard pass-through security, with all the (prepayment)
risk that it embeds.
What is the price impact of changes in PSA to the PAC Tranche and Companion
Tranche? By construction, so long as the PSA level is within the boundaries and
the Companion Tranche is not retired, the PAC Tranche cash flow is given by its
scheduled cash flow. As such, the price does not change at all. For instance, the
price of the PAC Tranche corresponding to Panels A, B, and C in Figure 8.10 is
$377.16 million. That is, moving the PSA between 100% and 300% does not change
314
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
Figure 8.9
PAC Scheduled Cash Flow
12
80% PSA
300 % PSA
Planned Cash Flow
10
Cash Flows
8
6
4
2
0
0
5
10
15
Years
20
25
30
the value at all. In this sense, this tranche is not subject to prepayment risk. Since
the pass through security is affected by changes in PSA, it must be the case that the
Companion Tranche value changes substantially with PSA. Indeed, the value of the
Companion Tranche in the three panels is $269.77, $257.62, and $250.54 million,
respectively. This variation induces a strong negative convexity effect. This negative
relation between interest rates and the Companion Tranche is evident in Panel A of
Figure 8.11. This figure reports the result of the same exercise carried out in Table 8.4
and Figure 8.6. Namely, as the interest rate declines, the PSA is increased accordingly
to capture the increase in prepayment that lower interest rates would trigger. The
PAC Tranche displays essentially a linear relation with interest rates, except for low
interest rates. From Table 8.4 we assume that PSA reaches all the way to 500%,
which is much larger than the upper PSA amount of 300%. We know that in this
case, the PAC reverts back to a pass through, and hence the negative convexity. The
Companion Tranche displays a much stronger negative convexity than the original
Pass Through, with its valuation becoming almost constant as interest rates decline.
8.4.3
Interest Only and Principal Only Strips.
Interest only and principal only strips constitute perhaps the simplest way to cut up the cash
flow from the original pass-through security.
• Interest-Only (IO) strips receive all of the interest payment from the underlying
collateral and none from the principal. So, for instance, if interest rates decline and
COLLATERALIZED MORTGAGE OBLIGATIONS
Figure 8.10
315
PAC Scheduled Cash Flow and True PSA
100 % PSA
200 % PSA
7
10
6
8
Cash Flow
Cash Flow
5
4
3
6
4
2
2
1
0
0
10
20
Years
300 % PSA
12
0
30
0
Total Pass−Through CF
PAC Tranche CF
Companion Tranche CF
14
30
20
30
400 % PSA
10
8
Cash Flow
Cash Flow
20
Years
12
10
6
4
8
6
4
2
0
10
2
0
10
20
Years
30
0
0
10
Years
316
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
Figure 8.11 PAC and Companion Tranche Value
450
300
Companion Tranche
250
PAC Tranche A
350
200
300
250
Companion Tranche Value
PAC Tranche Value
400
150
2
3
4
5
6
Interest Rate (%)
7
8
9
100
many homeowners prepay their debt, lower interest payments will occur in the future
as now less principal is available to compute the interest. As a consequence, an
investor who is long the IO will receive lower future cash payments (and the IO loses
value).
• Principal-Only (PO) strips receive all of the principal payments, both scheduled and
unscheduled, from the underlying principal, and no interest. So, for instance, if
homeowners prepay their debt because interest rates decline, all of the prepayment
goes to the PO strips. As a consequence, an investor who is long the PO will receive
a higher cash flow in the immediate future. A decline in interest rates therefore has
a large impact on the price of the PO.
IOs and POs have some additional peculiarities that are worth illuminating in the next
example.
EXAMPLE 8.8
Consider once again the pass through in Example 8.2 and the computations in Table
8.3. The payments to the IO strip simply correspond to the interest payments for
the pass through, contained in Column 8 of Table 8.3. The payments to the PO strip
correspond to the sum of Columns 6 and 7 of the same table. The interesting part
about IOs and POs is their behavior as the speed of prepayment changes. Table 8.7
shows the impact that lower interest rates and higher PSA have on the value of the
strips. The most interesting effect is the fact that IO value decreases substantially
as interest rates decline and PSA increases. In contrast, to counterbalance, the PO
strip must increase dramatically to ensure a higher total value, as shown earlier in
Table 8.3 for the pass-through security. This behavior greatly affects the effective
SUMMARY
317
Table 8.7 Value and Duration of IO and PO Strips
Interest Rate
PSA
IO Value
PO Value
2.00%
2.50%
3.00%
3.50%
4.00%
4.50%
5.00%
5.50%
6.00%
6.50%
7.00%
7.50%
8.00%
8.50%
9.00%
500
450
400
350
300
250
200
150
100
90
80
70
60
50
40
131.75
140.05
149.71
161.06
174.57
190.86
210.78
235.52
266.75
267.07
267.35
267.59
267.79
267.95
268.07
556.05
541.58
525.05
505.91
483.42
456.59
423.98
383.61
332.58
309.89
287.18
264.47
241.76
219.06
196.38
duration of IO and PO strips. Indeed, consider again the case in which the current
rate is r = 5%, and PSA = 200%. By applying the formula in Equation 8.18 we can
compute the effective duration of IO and PO strips as follows:
DI O
DP O
1 235.52 − 190.86
1 P I O (+50 bps) − P I O (−50 bps)
=−
I
O
P
2 × (50 bps)
210.78 2 × (50 bps)
= −21.19
1 383.61 − 456.59
1 P P O (+50 bps) − P P O (−50 bps)
=−
≈ − PO
P
2 × (50 bps)
423.98 2 × (50 bps)
= 17.21
≈
−
The effective duration of the IO strips is negative, and substantial, consistently with
the previous discussion. In contrast, the effective duration of the PO strips is strongly
positive. This is not surprising, as the weighted average of the effective durations of
the IO and PO strips must add up to the effective duration of the original pass-through
security, which we found to be 4.46 in Example 8.3. Thus, if the IO strips have a large
negative duration, it must be the case that PO strips have a large positive duration.
8.5
SUMMARY
In this chapter we covered the following topics:
1. Securitization: Securitization is the process through which financial institutions pool
similar assets in a portfolio and sell the portfolio to investors. Investors purchasing
the portfolio of assets receive claims on the cash flows generated by the assets in the
portfolio. Such assets are the collateral of the portfolio. If such assets are residential
318
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
mortgages, then the claims the investors obtain are called residential mortgage backed
securities. These assets can also be commercial mortgages, credit card receivables,
auto loans, and so on.
2. Fixed Rate Mortgages: In a fixed rate mortgage, the monthly mortgage payment
contains two components: Principal payment and interest rate payment. The interest
rate payment depends on the outstanding balance, and thus it is initially high and
declines over time.
3. Prepayment option: Homeowners have the right to prepay their mortgages, which
they tend to do when interest rates decline. Prepaying the mortgage implies that
banks receive their money earlier than expected, and can only reinvest at now lower
interest rates.
4. CPR and PSA: These are measures of prepayment speed. CPR assumes a constant
prepayment rate over the life of the mortgage backed security, while PSA assumes
lower prepayment for young pools of mortgages.
5. Pass-Through Mortgage Backed Securities: This securities are collateralized by
pools of mortgages, which are claim to the cash flows originating from the pool of
mortgages. In particular, they offer no protection against prepayments.
6. Negative Convexity: Although like any other bond, the value of a mortgage and thus
of pools of mortgages increases when interest rates decline, the value increases at a
declining rate, as homeowners repay their mortgages.
7. Collateralized Mortgage Obligations (CMOs): CMOs are structured securities that
offer various cash flow profiles to investors in MBS. Collateralized by an underlying
pass through security, CMOs typically allocate principal payments to various classes
according to specific formulas. For instance, Sequential Tranches receive principal
payments sequentially over time. Similarly, PAC Tranches receive principal payments according to a specific formula, while any discrepancy from the formula is
channeled to a Support tranche.
8.6 EXERCISES
1. Mortgage backed securities make coupon payments on a monthly basis. This means
that the yield curve should be estimated in this frequency. Unfortunately, Treasuries
traded on a given day are not enough in order to use the standard bootstrap techniques
to obtain the monthly yield curve, as the maturity usually spans for up to 30 years.
Other techniques such as the Extended Nelson Siegel model discussed in Chapter 2
should be used instead to obtain the yield curve. Given the data on bonds in Table
8.8 traded on December 1, 2000, compute the monthly yield curve and the discounts
for the next 30 years.
2. Consider the following MBS pass through with principal $300 million. The original
mortgage pool has a WAM = 360 months (30 years) and a WAC = 7.00%. The
pass through security pays a coupon equal to 6.5%. Instead of the yield curve,
you are given the following parameters from the extended Nelson Siegel model (see
EXERCISES
319
Table 8.8 Treasury Securities on December 1, 2000
Coupon
Maturity
Mid
Coupon
Maturity
Mid
Coupon
Maturity
Mid
0.000
0.000
0.000
0.000
0.000
0.000
4.625
0.000
0.000
0.000
0.000
4.500
0.000
0.000
0.000
5.000
0.000
0.000
0.000
0.000
0.000
4.875
0.000
0.000
0.000
0.000
5.000
0.000
0.000
5.625
0.000
0.000
0.000
0.000
5.750
5.500
7.875
0.000
5.500
5.625
5.875
7.500
0.000
12/7/2000
12/14/2000
12/15/2000
12/18/2000
12/21/2000
12/28/2000
12/31/2000
1/4/2001
1/11/2001
1/18/2001
1/25/2001
1/31/2001
2/8/2001
2/15/2001
2/22/2001
2/28/2001
3/1/2001
3/8/2001
3/15/2001
3/22/2001
3/29/2001
3/31/2001
4/5/2001
4/12/2001
4/19/2001
4/26/2001
4/30/2001
5/3/2001
5/10/2001
5/15/2001
5/17/2001
5/24/2001
5/31/2001
6/7/2001
6/30/2001
7/31/2001
8/15/2001
8/30/2001
8/31/2001
9/30/2001
10/31/2001
11/15/2001
11/29/2001
99.898
99.784
99.748
99.699
99.646
99.567
99.832
99.438
99.316
99.200
99.081
99.707
98.843
98.726
98.608
99.703
98.490
98.381
98.265
98.151
98.035
99.535
97.911
97.804
97.686
97.569
99.520
97.471
97.359
99.750
97.245
97.142
97.032
96.922
99.723
99.555
101.133
95.663
99.504
99.563
99.789
101.262
94.368
5.875
6.125
6.250
14.250
6.250
6.500
6.375
7.500
6.500
6.250
6.000
6.375
6.125
5.875
5.750
11.625
5.625
5.625
5.500
6.250
5.500
5.500
5.750
10.750
5.500
5.375
5.250
4.250
4.750
5.250
6.000
5.875
7.500
6.500
6.500
5.750
5.625
6.875
7.000
6.500
6.250
6.625
6.125
11/30/2001
12/31/2001
1/31/2002
2/15/2002
2/28/2002
3/31/2002
4/30/2002
5/15/2002
5/31/2002
6/30/2002
7/31/2002
8/15/2002
8/31/2002
9/30/2002
10/31/2002
11/15/2002
11/30/2002
12/31/2002
1/31/2003
2/15/2003
2/28/2003
3/31/2003
4/30/2003
5/15/2003
5/31/2003
6/30/2003
8/15/2003
11/15/2003
2/15/2004
5/15/2004
8/15/2004
11/15/2004
2/15/2005
5/15/2005
8/15/2005
11/15/2005
2/15/2006
5/15/2006
7/15/2006
10/15/2006
2/15/2007
5/15/2007
8/15/2007
99.801
100.066
100.250
109.434
100.316
100.707
100.684
102.277
100.941
100.664
100.402
101.008
100.625
100.301
100.148
110.707
100.031
99.992
99.727
101.207
99.742
99.723
100.273
111.461
99.742
99.461
99.148
96.355
97.527
98.969
101.434
101.086
107.125
103.867
103.930
101.344
100.324
106.098
106.750
104.563
103.555
105.723
103.180
5.500
5.625
4.750
5.500
6.000
6.500
5.750
11.250
10.625
9.875
9.250
7.250
7.500
8.750
8.875
9.125
9.000
8.875
8.125
8.500
8.750
8.750
7.875
8.125
8.125
8.000
7.250
7.625
7.125
6.250
7.500
7.625
6.875
6.000
6.750
6.500
6.625
6.375
6.125
5.500
5.250
5.250
6.125
2/15/2008
5/15/2008
11/15/2008
5/15/2009
8/15/2009
2/15/2010
8/15/2010
2/15/2015
8/15/2015
11/15/2015
2/15/2016
5/15/2016
11/15/2016
5/15/2017
8/15/2017
5/15/2018
11/15/2018
2/15/2019
8/15/2019
2/15/2020
5/15/2020
8/15/2020
2/15/2021
5/15/2021
8/15/2021
11/15/2021
8/15/2022
11/15/2022
2/15/2023
8/15/2023
11/15/2024
2/15/2025
8/15/2025
2/15/2026
8/15/2026
11/15/2026
2/15/2027
8/15/2027
11/15/2027
8/15/2028
11/15/2028
2/15/2029
8/15/2029
99.676
100.457
94.941
99.586
102.883
106.598
101.734
152.938
147.813
140.813
134.875
114.938
117.719
131.156
132.688
136.250
135.438
134.281
126.281
130.973
134.039
134.281
124.375
127.500
127.656
126.391
117.688
122.406
116.313
105.531
121.844
123.594
114.063
102.719
112.750
109.531
111.281
108.094
104.785
96.574
93.250
93.402
105.969
c 2009 Center for Research in Security Prices (CRSP),
Source: Data excerpted from CRSP (Daily Treasuries) The University of Chicago Booth School of Business.
320
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
Table 8.9 Tranches of a 6.5% Pass-Through Security
Tranche
Principal
Interest
175
75
30
20
6.50%
6.50%
6.50%
6.50%
Tranche A
Tranche B
Tranche C
Tranche D
Chapter 2): θ0 = 6, 278.30, θ1 = −6, 278.25, θ2 = −6, 292.47, θ3 = 0.04387,
λ1 = 27, 056.49, and λ2 = 30.48. That is, to compute the continuously compounded
zero coupon yield with maturity T , the formula is
1 − e− λ 1
T
r(0, T )
= θ0 + (θ1 + θ2 )
T
λ1
1 − e− λ 2
T
− θ2 e− λ 1 + θ3
T
T
λ2
− e− λ 2
T
(8.23)
The discount with maturity T is then Z(0, T ) = e−r (0,T )×T .
(a) What is the price of the pass through? Assume a constant PSA = 150%.
(b) Compute the duration of this security assuming that the PSA remains constant
at 150%.
(c) Compute the effective duration of this security assuming that the PSA increases
to 200% if the term structure shifts down by 50 basis points, while it decreases
to 120% if the term structure shifts up by 50 basis points. Comment on any
difference compared to your result in part (b).
(d) Compute the effective convexity of this security under the same PSA assumptions as in part (c). Interpret your results.
3. Consider the following MBS pass through with principal $300 million. The original
mortgage pool has a WAM = 360 months (30 years) and a WAC = 7.00%. The
pass-through security pays a coupon equal to 6.5%. Use the same spot rate r(0, T )
as computed in Exercise 2 for your calculations. This security is divided into four
tranches (A,B,C, and D), each with the principal in Table 8.9:
(a) What is the price of each tranche? Assume a constant PSA = 150%.
(b) Compute the effective duration of the various tranches assuming that the PSA
increases to 200% if the term structure shifts down by 50 basis points, while
it decreases to 120% if the term structure shifts up by 50 basis points. Which
tranche is more sensitive to interest rate movements? Which tranche is less
sensitive?
(c) Compute the effective convexity of the various tranches under the same PSA
assumptions as in part (b). Interpret your results.
(d) If you decide to buy all the tranches, is this the same as holding the MBS pass
through in Exercise 2? (e.g., Does it have the same price? Same duration?)
EXERCISES
Table 8.10
Tranche
Tranche A
Tranche B
Tranche C
Tranche Z
321
Tranches of a 6.5% Pass-Through Security
Principal
Interest
175
75
30
20
6.50%
6.50%
6.50%
6.50% (accrual)
4. Consider the following MBS pass through with principal $300 million. The original
mortgage pool has a WAM = 360 months (30 years) and a WAC = 7.00%. The
pass-through security pays a coupon equal to 6.5%. Use the same spot rate r(0, T )
as computed in Exercise 2 for your calculations. This security is divided into four
tranches (A,B,C, and Z), each with the principal in Table 8.10. Tranche Z is an
accrual tranche that does not receive payments until all other tranches are paid off.
This means that interest payments are accrued (nominally added to the principal of
the tranche) until the other tranches’ principal is fully paid.
(a) What is the price of each tranche? Assume a constant PSA = 150%.
(b) Compute the effective duration of the various tranches assuming that the PSA
increases to 200% if the term structure shifts down by 50 basis points, while
it decreases to 120% if the term structure shifts up by 50 basis points. Which
tranche is more sensitive to interest rate movements? Which tranche is less
sensitive?
(c) Is Tranche Z more sensitive to interest rate changes than Tranche D from
Exercise 4? What about tranches A, B, and C? Are they more sensitive to
interest rate changes if they are supported by Tranche D or by Tranche Z?
(d) Compute the effective convexity of the various tranches under the same PSA
assumptions as in part (b). Interpret your results.
(e) If you decide to buy all the tranches, is this the same as holding the MBS pass
through from Exercise 2 (e.g. Does it have the same price? Same duration?).
5. Consider the following MBS pass through with principal $300 million. The original
mortgage pool has a WAM = 360 months (30 years) and a WAC = 7.00%. The pass
through security pays a coupon equal to 6.5%. Use the same spot rate r(0, T ) as
computed in Exercise 2 for your calculations. The security is divided into a Plan
Amortization Class (PAC) and a support tranche according to Table 8.11.
The PAC has an upper collar of 300% PSA and a lower collar of 85% PSA.
(a) What is the price of each tranche? Assume a constant PSA = 150%.
(b) Compute the effective duration of the two tranches assuming that the PSA
increases to 200% if the term structure shifts down by 50 basis points, while
it decreases to 120% if the term structure shifts up by 50 basis points. Which
tranche is more sensitive to interest rate movements? Which tranche is less
sensitive?
322
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
Table 8.11 Tranches of 6.5% Pass Through Security
Tranche
Principal
Interest
PAC
Support
181.34
118.66
6.50%
6.50%
(c) Compute the effective convexity of the various tranches under the same PSA
assumptions as in part (b). Interpret your results.
(d) If you decide to buy all the tranches, is this the same as holding the MBS pass
through from Exercise 3 (e.g., Does it have the same price? Same duration?).
6. Consider the following MBS pass through with principal $300 million. The original
mortgage pool has a WAM = 360 months (30 years) and a WAC = 7.00%. The
pass-through security pays a coupon equal to 6.5%. Use the same spot rate r(0, T )
as computed in Exercise 2 for your calculations. The pass through is divided into a
Principal Only tranche and an Interest Only tranche.
(a) What is the price of each tranche? Assume a constant PSA = 150%.
(b) Compute the effective duration of the IO and PO tranches assuming that the
PSA increases to 200% if the term structure shifts down by 50 basis points,
while it decreases to 120% if the term structure shifts up by 50 basis points.
Which tranche is more sensitive to interest rate movements? Which tranche is
less sensitive?
(c) Compute the effective convexity of the IO and PO tranches under the same PSA
assumptions as in part (b). Interpret your results.
(d) If you decide to buy all the tranches, is this the same as holding the MBS pass
through from Exercise 2 (e.g., Does it have the same price? Same duration?).
7. The following exercise is based on a series of investments made in 1993 by City
Colleges of Chicago (CCC), a system of community colleges. Its treasurer decided
to invest up to 70% of its portfolio in the lower tranches of a Fannie Mae MBS:
FNMA 1993-237.15 All payments within this trust were Principal-Only (PO); this
particular type of security was called a Stripped Mortgage Backed Security (SMBS).
The FNMA 1993-237 had a principal balance of $425 million with a WAM = 348 and
WAC = 8.27%. Because all tranches were PO, the coupon rate of the underlying pass
through is not needed. The security was divided in the tranches and types reported
in Table 8.12, where PAC stands for Planned Amortization Class, TAC stands for
Targeted Amortization Class, and SUP stands for Support Class.
In Table 8.12, UC and LC stand for the upper collar and the lower collar for the PACs.
TAC are similar to PAC but, instead of using a range of PSA, use only a single value
15 Information
for this case was obtained from documents rendered by the United States Court of Appeals for the
Fifth Circuit: Westcap Corp. vs. City Colleges of Chicago (CCC) available in Lexis (25502). CCC also invested
in another similar security: FNMA 1993-205, which we omitted to simplify the exercise.
EXERCISES
Table 8.12
323
Tranches in FNMA, 1993-237
Tranche
Principal
Type
UC
LC
A
B
C
E
G
H
127.50
51.00
25.50
68.00
59.50
93.50
PAC
PAC
PAC
TAC
TAC
SUP
550%
550%
550%
300%
450%
135%
135%
135%
Total
425.00
of PSA to create a schedule of payments. The prospectus for the security pointed out
the following principal distribution plan:
Principal will be distributed monthly on the Certificates in an amount (the “Principal Distribution Amount”) equal to the aggregate distributions of principal concurrently made on the SMBS. On each Distribution Date, the Principal Distribution
Amount will be distributed as principal of the Classes in the following order of
priority:
(a) sequentially, to the A, B and C Classes [Planned Amortization Class (PAC)],
in that order, until the principal balances thereof are reduced to their respective Planned Balances for such Distribution Date;
(b) sequentially, to the E and G Classes [Targeted Amortization Class (TAC)], in
that order, until the principal balances thereof are reduced to their respective
Targeted Balances for such Distribution Date;
(c) to the H Class [Support Class], until the principal balance thereof is reduced
to zero;
(d) to the G and E Classes, in that order, without regard to their Targeted Balances
and until the principal balances thereof are reduced to zero;
(e) to the A Class, without regard to its Planned Balance and until the principal
balance thereof is reduced to zero; and
(f) concurrently, to the B and C Classes, in proportion to their then current
principal balances, without regard to the Planned Balances and until the
principal balances thereof are reduced to zero.
On October 1, 1993 the market faced the yield curve summarized by the following Extended Nelson Siegel model parameters (see Chapter 2 and Equation 8.23
above): θ0 = 6, 278.30, θ1 = −6, 278.30, θ2 = −6, 291.28, θ3 = 0.70906,
λ1 = 27, 056.50, and λ2 = 20.2312.
(a) According to industry experts you find out that the PSA is currently at 450%.
i. Assuming a constant PSA rate, value the tranches of FNMA 1993-237.
ii. What is the duration of each of the tranches? Are G and H the tranches
with the highest duration?
(b) CCC decides to invest $100 million, divided equally into tranches G and H.
After 6 months you receive new data to compute the yield curve (it is now April
324
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
4, 1994) according to the Extended Nelson Siegel model: θ0 = 6, 278.30,
θ1 = −6, 278.30, θ2 = −6, 291.28, θ3 = 0.97584, λ1 = 27, 056.50, and
λ2 = 20.2249.
i. Have interest rates risen? Stayed the same? Or fallen?
ii. What will the value of the portfolio be on April 4, 1994 assuming that the
PSA stays the same?
(c) You find out that on April 4, 1994 the PSA is actually 200%.
i. Is the change in the PSA a reflection of what happens to the yield curve?
Why or why not?
ii. Will this change offset the P/L in the portfolio’s value from the change in
interest rates?
iii. Compute the price of each tranche at April 4, 1994. Do the changes reflect
what you expected from the duration calculation?
iv. What is the gain / loss of the portfolio at this time?
8.7 CASE STUDY: PiVe INVESTMENT GROUP AND THE HEDGING OF
PASS-THROUGH SECURITIES
It is June 8, 2007, and the principals of PiVe Investment Group are considering how to best
hedge their recent investment of $300 million in pass through securities. In particular, they
just invested in the Ginnie Mae I pass through security, GNSF 6, in the TBA market, which
was trading at a discount. The underlying pool of this pass-through security has a WAC
of 6.5% – very close to the current mortgage rate for new 30-year fixed-rate mortgages of
6.59% – and a WAM of 320 months. The pass through security has a coupon of 6%, and it
is now quoted in the TBA market at bid/ask prices = [99.4375 / 99.40625].
The underlying decision to purchase this pass-through security was based on its substantial spread over Treasuries compared to the security PSA level. The principals of PiVe
Investment Group noticed that the median “street” forecast of the long-term PSA level from
SIFMA (Securities Industry and Financial Markets Association) was only a 225% PSA,
and forecasts ranged between 172 and 304 (see Table 8.13).16 The current term structure
of interest rates was almost flat around the 5% rate, as the short-term rates were around
5% while the long-term rates were around 5.4%. Given the term structure of interest rates,
the principals obtained a value of the GNSF 6 of P = 104.05 at the median 225% PSA,
much higher than the traded price. Other PSA scenarios also generated higher prices than
the traded price. In particular, they obtained P = 104.63 at 172% PSA and P = 103.38 at
304% PSA. PiVe Investment Group used to gauge the attractiveness of their investments by
computing the implied spread over Treasuries that justified the traded price. In this case,
a uniform continuously compounded spread of 1.015% over all maturities gave a price of
P = 99.43, which is within the bid/ask prices.
PiVe principals were very aware of the source of the price discount: The negative
convexity induced by prepayment risk. Since the WAC was close to the current mortgage
rate, a further decline in interest rates might spur a large refinincing wave, such as the one
16 Information
about SIFMA prepayment tables is available at the SIFMA Web site www.sifma.org/research/
statistics/mbs prepayment.html.
CASE STUDY: PiVe INVESTMENT GROUP AND THE HEDGING OF PASS-THROUGH SECURITIES
325
experienced in 2001 - 2002, implying that the price would flatten out at around 100 instead
of increasing in value, as other Treasury securities would. Moreover, this negative convexity
tends to generate trading losses in average, which conterbalance the higher yield.17 Still,
the 1% spread over Treasuries seemed to be a large compensation for such prepayment
risk, and this judgment spurred the purchase of this particular security. The principals are
now deciding how to hedge away interest rate risk.
8.7.1
Three Measures of Duration and Convexity
The principals of PiVe Investment Group are looking at three different measures of duration,
which are not exactly identical to each other, and they must consider which one to use to
set up an effective hedging strategy against the movement of interest rates.
8.7.1.1 Simple Duration Given the current median market forecast of the PSA level
of 240%, the principals of PiVe computed first a benchmark value of duration. Keeping
the PSA constant, they can compute the duration in the usual way as for Treasuries. In
particular, it is simple to see that from the definition of duration, we obtain
D
C
= −
=
1 dP
=
P dr
1 d2 P
=
P d r2
320
wi × Ti
i=1
320
wi × Ti2
i=1
where Ti are the monthly payment times, and the weights are
wi =
Ti )
CFi × Z(0,
P
Ti ) represents
In this formula, the CFi are computed from the projected PSA level, and Z(0,
the discount obtained by adding the 1.03% spread over Treasuries that is needed to match
the price from the constant PSA to the traded price. This calculation implies a duration
equal to D = 4.39 and C = 36.1590.
The principals of PiVe capitals were skeptical about this computation, though. While
the duration figure appears reasonable, the convexity figure appears wrong, as it implies a
positive convexity instead of a negative one. This doubt prompted PiVe principals to search
for alternative methodologies.
8.7.2
PSA-Adjusted Effective Duration and Convexity
PiVe principals recognized an important shortcoming of the earlier computations, and this
was the fact that the PSA is not allowed to change with the interest rate. The second
methodology corrects the shortcoming as it takes into account changes in the PSA that
are induced by variations in interest rates. In particular, the first step in computing the
PSA-adjusted duration and convexity is to estimate the change in the PSA level that is
induced by a change in interest rates. Luckily, SIFMA projection tables contain the median
17 See
discussion in Chapter 4.
326
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
“street” forecast of PSA under various scenarios about the change in interest rates. These
projections are contained in Table 8.13. For instance, according to these projections, a
decrease in the yield curve of 50 basis points would result in an increase in the PSA from
225% to 312%, while an increase in the yield curve by the same amount would decrease
the PSA from 225% to 179%. The principals of PiVe reasoned that they could compute
the value of the security under the two scenarios of an increase and decrease in interest
rates while taking into account the predicted change in the PSA, and thus obtain a better
measure of duration and convexity. These are given by the effective duration and convexity
formulas in Equations 8.18 and 8.19, respectively:
1 P (+50bps) − P (−50bps)
P
2 × 50bps
1 P (+50bps) + P (−50bps) − 2 × P
P
(50bps)2
D
≈ −
C
≈
where P is the current traded price, and P (+50bps) and P (−50bps) are the pass-through
security prices under the two scenarios of an increase and decrease in the yield by 50 basis
points, respectively. PiVe obtained P (−50bps) = 101.43, where the calculation also used
the market forecast P SA = 312, and P (+50bps) = 96.79, where the calculation also used
P SA = 179. Substituting these numbers in the formulas above, PiVe principals obtained
a duration equal to D ≈ 4.68 and a convexity C ≈ −251.08. While the duration is only a
slightly higher value than the benchmark duration – probably due to the large decrease in
the PSA if the interest rate increases – the convexity measure is completely different from
the benchmark value, and it now conforms to intuition about prepayment risk.
To check their computations, PiVe principals also computed the approximate duration
and convexity implied by a large change of interest rates, of 100 bps:
D
C
1 94.00 − 102.24
1 P (+100bps) − P (−100bps)
=−
P
2 × 100bps
99.43 2 × 100bps
= 4.14
1 P (+100bps) + P (−100bps) − 2 × P
1 94.00 + 102.24 − 2 × 99.43
≈
=
P
(100bps)2
99.43
(100bps)2
= −262.63
≈
−
In this case, the duration approximation is slightly smaller than the benchmark computation,
but again the convexity number is negative, and similar to the one they obtained previously.
8.7.3 Empirical Estimate of Duration and Convexity
As the principals of PiVe Investment Group were discussing these issues and reading the
research literature on hedging interest rate risk, they were struck by a figure they saw in
a recent academic paper, soon to be published in the Review of Financial Studies. In this
article, written by Professors Jefferson Duarte, Francis Longstaff and Fan Yu,18 Figure 3
plots the price of the 30-year Ginnie Mae I against the five year swap rate. It appears
from the figure that there is a strong relation between these two variables. The principals
18 See
Duarte, Longstaff, and Yu (2007). This section presents a much simplified version of their more rigorous
approach. See the paper for details.
CASE STUDY: PiVe INVESTMENT GROUP AND THE HEDGING OF PASS-THROUGH SECURITIES
Table 8.13
327
Mortgage Prepayment Projection
GNMA I 30 Y
Participating dealers: BS CITI CTW GC GS JPM LB ML MS UBS
Yield Curve Scenarios
Avg -300 Avg -200 Avg -100 Avg -50 Avg Base Avg +50 Avg +100 Avg +200 Avg +300 Low - High
983
854
508
312
225
179
156
130
116
172–304
Source: SIFMA.
of PiVe immediately gathered some data, and found that indeed there is a strong relation:
The results of their analysis is in Figure 8.12. Panel A reports the time series of both the
the quoted TBA price of GNSF 6 and the 5-year swap rate from June, 1995 to May, 2007.
The two series are almost the mirror image of each other, although it appears that when
the swap rate declines the price of GNSF 6 does not increase as much as it declines when
the swap rate increases. This asymmetry is due to the negative convexity of GNSF 6 with
respect to interest rate. Panel B highlights this negative convexity, as the figure contains a
scatter plot of the two series in Panel A. Panel B also plots the fitted value of the simple
fourth order polynomial regression:
4
Pt = α +
β i × ct (5)i + t
i= 1
where Pt is the GNSF price and ct (5) the 5-year swap rate at time t. The fitted value is
presented as the bold line in the figure. The R2 of this regression is 94.3%.19
This empirical estimate of the relation between GNSF 6 prices and the swap rate also
allows the PiVe principals to compute its duration and convexity. In particular, given the
fitted value of the regression, they can compute the first and second derivative of GNSF 6
price P with respect to the 5-year swap rate:20
dP
d c(5)
d2 P
d c(5)2
= β 1 + 2 × β 2 × c(5) + 3 × β 3 × c(5)2 + 4 × β 4 × c(5)3
=
2 × β 2 + 6 × β 3 × c(5) + 12 × β 4 × c(5)2
Given the current swap rate of c(5) = 5.61%, these formulas imply
and
2
d P
d c(5) 2
(8.24)
(8.25)
d P
d c(5)
= −444.66
= −14731.4. Because a parallel shift in the term structure induces an equal
estimated coefficients are α = 116.8853, β 1 = −1.2070e + 003, β 2 = 4.5123e + 004, β 3 =
−7.4270e + 005, and β 4 = 3.8397e + 006. They are significant with simple standard errors. However, given
the persistence of the two time series, a Newey West correction of standad errors shows that parameters are not
significant. The high R 2 , the flip/flow sign of the regression coefficients and their lack of statistical significance
is evidence of multicollinearity, typical in polynomial regressions. See Duarte, Longstaff and Yu (2007) for a
more advanced estimation methodology that does not suffer from these problems.
20 The result follows from the rules of the first and second derivative of a function F (x) = x i with respect to x.
These are F (x) = i × xi −1 and F (x) = i × (i − 1) × xi −2 .
19 The
328
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
increase in the swap rate,21 we obtain the following estimates of the duration and convexity
Empirical Duration
Empirical Convexity
444.66
1 dP
=
= 4.4725
P d c(5)
99.4219
−14731.4
1 d2 P
= −148.17
=
P d c(5)2
99.4219
= −
(8.26)
=
(8.27)
The duration is once again similar to the previously computed values providing additional
confidence to the PiVe principals about the correct hedge ratio. The convexity, while
negative, is smaller than the one computed using PSA projections. The PiVe principals
conjectured that one explanation of the difference may be that while the empirical variation
reflects an historical average about the relation between prices and swap rates, the PSA
projections may include more recent market trends, that may have increased the convexity
of GNSF 6 compared to swap rates. Another possibility is that the polynomial regression
is rather restrictive in determining the exact shape of GNSF 6 price with respect to swap
rates.
8.7.4 The Hedge Ratio
Given an estimate of the duration D of GNSF 6, what is the position in the 5-year swaps
that PiVe capital has to take? First, an increase in the interest rate causes the value of GNSF
to drop. To counterbalance this change, we need a position N in a swap that increases in
value when the interest rate drops. That is, we must enter the swap as a fixed-rate payer.
Thus, the change in the portfolio Π that is long the Ginnie Mae I and a fixed rate swap has
the following sensitivity to changes in the interest rate
dP
d V sw ap (5)
dΠ
=
+N ×
= −D × P + N −DS$ w ap
dr
dr
dr
sw ap
(5) is the value of the 5-year swap, N is the notional position in the swap, and
where V
DS$ w ap is the swap dollar duration. From Chapter 5, a swap is given by a portfolio that is long
a floating rate bond (with duration close to zero) and short a fixed rate coupon bond. From
the current swap curve, PiVe principals compute the duration of the fixed-rate leg of the
w ap
w ap $
= 4.3934, and its dollar duration per $100 notional as DfSixed
= 439.34.
swap as DfSixed
w ap
Given that the dollar duration of the floating rate bond is just DfSloat
w ap $
DfSloat
$
= 25, the swap
w ap $
DfSixed
=
−
= 414.34 for $100 notional.
dollar duration is D
The hedging strategy requires the portfolio Π to be insensitive to changes in interest
rates. That is, we want to find N such that
S w ap$
dΠ
=0
dr
This condition implies that the notional is given by:
N =−
D×P
4.4725 × $300 million
= 3.23 million
=
$
414.3369
DS w ap
where $300 million is the position in GNSF 6. Because the dollar duration of the swap was
on a $100 notional, we have that the notional in the 5-year swap must be $323 million.
21 This statement is exactly true if the term structure of interest rates is flat,
to the yield-to-maturity, instead of the spot rate curve.
or if duration is computed with respect
329
CASE STUDY: PiVe INVESTMENT GROUP AND THE HEDGING OF PASS-THROUGH SECURITIES
Figure 8.12
GNSF 6 Price and the 5-Year Swap Rate
Panel A: Time Series
0.1
5−year
Swap Rate
100
0.05
GNSF 6
Price
80
1994
1996
1998
2000
2002
2004
2006
0
2008
Panel B: Scatterplot
110
Data
Fitted Value
GNSF 6 Price
105
100
95
90
85
80
0.02
0.03
Data Source: Bloomberg.
0.04
0.05
0.06
5 Year Swap Rate
0.07
0.08
0.09
5 year Swap Rate
GNSF 6 Price
120
330
BASICS OF RESIDENTIAL MORTGAGE BACKED SECURITIES
8.8 APPENDIX: EFFECTIVE CONVEXITY
The approximating formula in Equation 8.19 stems from the definition of second derivative,
discussed in Chapter 4. Specifically, the second derivative equals the change in the first
derivative, i.e., the slope of the curve at the current interest rate:
d (dP/dr)
d2 P
.
=
dr2
dr
The slope of the curve at the current interest rate can be approximated by dP/dr ≈
(P − P (−x bps))/(x bps), while the slope of the curve when the interest rate increases by
x bps can be approximated as dP/dr ≈ (P (+x bps) − P )/(x bps). Taking the difference
in slopes, and dividing by x bps, gives Equation 8.19, that is,
P (+x bps)−P
P −P (−x bps)
−
x bps
x bps
1
C ≈
P
x bps
1 P (+x bps) + P (−x bps) − 2P
=
P
(x bps)2
PART II
TERM STRUCTURE MODELS:
TREES
In this second part of the book we move one step forward in the understanding of fixed
income instruments. In particular, we introduce the notion of no arbitrage, and the basics
of term structure modeling. To advance the topic, let’s consider the following example.
Example
Consider a trader in a prominent investment bank. By using a bootstrap methodology or
any of the other methodologies discussed in Chapter 2, the trader has estimated the current
discount function Z (0, T ). Recall that Z (0, T ) gives the value today (0) of one dollar at
time T .
If a client asks the trader to quote the price of 10%, 5-year, T-bond, the trader has all the
information needed. The price can be computed from
10
Z(0, Ti ) + 100 × Z(0, T )
P (0, T ) = 5 ×
i= 1
Suppose now that the client asks the trader to quote the price of a 10%, 5-year, callable
T-bond, that is, such that the Treasury has the option to buy it back at par at some date in
the future.
332
• How can the trader compute the value of such a bond?
• How can the trader effectively hedge it?
Methodology 1 (Naive)
Naively we can follow this reasoning:
1. We have data on interest rates, so we can use past data on interest rates to forecast
future interest rates.
2. The Treasury will exercise the options when interest rates are low in the future, as
low interest rates imply high bond prices.
3. By forecasting future interest rates, we can forecast the future cash flows of the bond:
If interest rates go up, the cash flow remains at 5% per period and 100 in 5 years. If
interest rates go down, the cash flow may stop early, as we will receive 100 as soon
as the Treasury calls the bond.
So, by using past interest rates, we can compute the expected cash flows for each maturity
T1 , .., Tn
E [CFT 1 ] , E [CFT 2 ] , ..., E [CFT n ]
At this point it appears we are done: We just need to compute the present value (PV) of
these cash flows
n
P C (0, T ) =
E [P V (CFT i )]
i= 1
However, here we stumble into a roadblock: How do we compute the present value? Can
we use again the discount function Z (0, T )?
No. The discount function Z (0, T ) is the the discount to be applied to future cash flows
that are known today. Instead, the cash flows from a callable bond are not known today,
and in fact, we can only estimate the expected cash flows.
One alternative would then be to use some sophisticated asset pricing model, such as
the Capital Asset Pricing Model, according to which we should discount cash flows that
are uncertain, and therefore possibly risky, by using a rate of return that is adjusted for a
risk premium. For instance, the Capital Asset Pricing Model implies computing the yield
of risky cash flows as y(t; T ) = r(0; T ) + β×(Market premium). However, even in this
case, what is the “beta” of these cash flows?
This methodology is very hard to implement, as it requires we are able to (a) forecast
future interest rates and thus compute expected cash flows; and (b) discount these expected
cash flows using a proper risk adjusted discount rate. Both are difficult, and imprecise
tasks. Term structure models are models for the interest rates that overcome both problems,
by using the notion of no arbitrage.
Methodology 2 (No Arbitrage)
A better methodology is to develop a model of the term structure, and use the concept
of no arbitrage to obtain the value of the callable bonds from the noncallable bond. More
specifically, the methodology that we use is the following:
333
1. Postulate a model for interest rates.
2. Estimate the model using bond data, so that the model is consistent with current
traded bonds.
3. Calculate the price of additional securities, such as the callable bonds, by using the
model with same estimated parameters.
Step 3 essentially involves pricing additional securities from more primitive securities,
using no arbitrage arguments.
The question then is how we can develop a model of the term structure. This has been
a very active field of research among academic researchers, traders, and practitioners in
investment banks and hedge funds. There are numerous models that have been proposed
for the term structure of interest rates, each of which has some good properties and some
bad properties. There isn’t such a thing as the perfect interest rate model (or the “right”
model), but only better and worse models of interest rates for the pricing of interest rate
securities. Some models have more realistic features than others, but they may be harder to
work with, or take longer to estimate, or to use to price fixed income securities. Some other
models may have less realistic features, but imply analytical formulas for many interest
rate derivatives, a property that make them particularly useful to value large portfolios of
securities. The choice of the proper model to use always depends on the task at hand.
In the coming chapters we discuss many models, and highlights their pros and cons. The
important concept across models, though, is the notion of no arbitrage, meaning that each
model has to be internally consistent and not leave any arbitrage opportunities on the table.
Traders then employ these models to both spot potential arbitrage opportunities, whenever
traded securities are not in line with the prediction of their models, or to price additional,
exotic securities from more primitive ones.
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CHAPTER 9
ONE STEP BINOMIAL TREES
In this chapter we introduce binomial trees. This is a model for interest rates that has
three advantages: (1) it is simple, as it does not require high-tech mathematics; (2) when
extended to multi-steps, it can be used to price most derivative securities; (3) it is used
extensively by practitioners to price real securities. Before we get to this latter part, we
must start from basics. We begin the analysis using a specific example, the term structure
of interest rates on January 8, 2002.1
9.1
A ONE-STEP INTEREST RATE BINOMIAL TREE
Today is January 8, 2002. The term structure of interest rates is depicted in Figure 9.1. The
prices and continously compounded yields of zero coupon bonds (STRIPS) up to T = 1.5
to maturity are provided in Table 9.1.
An interest rate model starts with the specification of the dynamics of the short-term
interest rate. These dynamics reflect our predictions of future interest rates, as discussed in
Chapter 7. We take the short-term interest rate as exogenous, in the sense that it is driven
by monetary policy choices, and market participants cannot affect it. In principle, this
characteristic is really appropriate for the overnight (Federal funds) rate, but for now we
keep matters simple, and we take the 6-month rate as the exogenous rate.
1 We
often use this date in Part II and III of the book, as the term structure on January 8, 2002 was very steep.
This fact generates interesting implications for term structure models.
335
336
ONE STEP BINOMIAL TREES
Figure 9.1
Term Structure of Interest Rates on January 8, 2002
5.5
5
4.5
Yields
4
3.5
3
2.5
2
1.5
0
1
2
3
4
5
6
Time−to−Maturity
7
8
9
Data Source: The Wall Street Journal.
Table 9.1 Interpolated Treasury STRIPS on January 8, 2002
Maturity
Price
Yield
0.5
1
1.5
99.1338
97.8925
96.1531
1.74 %
2.13 %
2.62 %
Data Source: The Wall Street Journal.
10
A ONE-STEP INTEREST RATE BINOMIAL TREE
Table 9.2
period =⇒
time (in years) =⇒
337
A One-Period Binomial Interest Rate Tree
i=0
t=0
i=1
t = 0.5
r1 , u = 3.39%
with prob. p = 1/2
r1 , d = 0.95%
with prob. 1 − p = 1/2
r0 = 1.74%
Given that the 6-month rate is much lower than in the past, we predict that it will rise
in the future. Suppose that our prediction of the interest rate in the future is in fact 2.21%.
Bear in mind that this is just a prediction, and we do not know for sure. In particular,
we believe that in six months there is an equal chance that the 6-month rate will be either
3.39% or 0.95%.2 We can graphically describe these beliefs in a binomial tree, as in Table
9.2.
In Table 9.2 r0 denotes the current continuously compounded 6-month interest rate,
which is given in the top row in Table 9.1. The notation r1,u indicates the 6-month (c.c.)
interest rate in period i = 1 after an up move (u) in the interest rate, while r1,d indicates
the 6-month interest rate at the same time after a down move (d). Given that each scenario
has a 50-50 chance, the expected (predicted) interest rate in six months is
E [r1 ] =
1
1
r1,u + r1,d = 2.17%
2
2
(9.1)
This binomial tree for interest rates produces interesting implications for the relative
pricing of Treasury bonds, as well as other securities whose value depends on interest rates.
The logic is the following: When we posit a model for interest rates such as the one in Table
9.2, we are implicitly imposing strong restrictions across bonds. In particular, it cannot be
the case that the 1-year bond and the 1.5-year bond move independently, for instance, as
both prices are going to be affected in a particular direction by the exogenous movement of
the short-term interest rate r1 . If r1 goes up, then both bonds will go down, and vice versa,
if r1 goes down, both the 1-year and the 1.5-year bond will go up in prices. This is because
the interest rate tree in Table 9.2 implicitly assumes that only the short term interest rate
matters for all of the bonds. As discussed in Chapter 7 this is indeed the hope of the Federal
Reserve: By changing only the (overnight) Federal funds rate, the Federal Reserve hopes
to be able to also influence long-term bonds. While in model 9.2 this happens, in richer
models we will see that the long-term bonds are affected by other factors as well. However,
for now, we keep things simple and find the no arbitrage relations that must exist between
bond prices.
2 These numbers are consistent with the volatility of interest rates being around 0.0173, as estimated from variation
√
√
of interest rates. Then, we have r1 , u = 0.0217 + 0.0173 × 0.5 and r1 , d = 0.0217 − 0.0173 × 0.5. These
numbers are obtained from a model introduced in Chapter 10
338
ONE STEP BINOMIAL TREES
9.1.1 Continuous Compounding
Note that we chose to model the binomial tree in Table 9.2 using continuously compounded
rates. An alternative is to consider semi-annually compounded interest rates on the tree,
since we are considering six-month intervals. However, our choice is motivated by the
following considerations: First, remember that there is always a one-to-one relation between
continuously compounded rates and any other rate, given by the fact that the following
relation must always hold
(9.2)
er ×Δ = 1 + rn × Δ
where Δ = 1/n is the compounding frequency, r is the continuously compounded rate, and
rn is the n−times compounded rate. Second, when we apply the model to the pricing of
realistic securities, we will consider very high frequency trees, easily with 200 or 300 steps.
Continuous compounding allows us to vary only the time interval between nodes, and not
also the rate itself as we increase the frequency of movement. Third, many interest rate
securities and derivatives have cash flows that depend on different compounding frequency:
for instance, simple plain vanilla swaps have floating rates compounded quarterly and fixed
payment compounded semi-annually. Using continuous compounding, and then Equation
9.2 allows for a simple method to model the cash flows of these securities. Finally, this
assumption allows us to better link the term structure trees developed in this part of the
book with the more advanced continuous time models developed in Part III.
9.1.2 The Binomial Tree for a Two-Period Zero Coupon Bond
The first implication of the interest tree in Table 9.2 together with the information in Table
9.1 is that we can obtain the 2-period tree for the bond with one year to maturity. First,
though, let’s introduce a convenient notation to denote the value of securities along the tree:
Pi,j (k) = Bond price in period i, in node j, and with maturity in period k
(9.3)
So for example P0 (2) denotes the zero coupon bond price at time 0, and maturing in period
k = 2 (i.e. time t = 1). Similarly, P1,u (2) denotes the zero coupon bond price in period
i = 1, if the interest rate went up – so in node u – and still maturing in period k = 2.
Moving now to the example, consider the bond that will mature in period i = 2, which
from Table 9.1 has a price of P0 (2) = 97.8925. If the interest rate goes up, to r1,u = 3.39%,
then the bond maturity in period i = 2 will have a price P1,u (2) = e−r 1 , u ×0.5 × 100 =
98.3193. If instead the interest rate moves down, to r1,d = 0.95%, the zero coupon bond is
P1,d (2) = e−r 1 , u ×0.5 × 100 = 99.5261. Since the zero coupon bond pays 100 at maturity
i = 2 independently of the behavior of the interest rate at that time, we obtain the tree in
Table 9.3.
9.2 NO ARBITRAGE ON A BINOMIAL TREE
We now exploit the binomial tree for the one-year zero coupon bond in Table 9.3 to obtain
the fair, no-arbitrage price of additional securities whose final payoff depends on the interest
rate.
Consider for instance an interest rate option with maturity i = 1, such that at time i = 1
pays the following amounts:
(Payoff at i = 1) = 100 × max (rK − r1 , 0)
(9.4)
NO ARBITRAGE ON A BINOMIAL TREE
Table 9.3
period =⇒
time (in years) =⇒
339
The One-Year Zero Coupon Bond Binomial Tree
i=0
t=0
i=1
t = 0.5
i=2
t=1
P 2 , u u (2) = 100
P 1 , u (2) = 98.3193
P 2 , u d (2)
= 100
P 2 , d u (2)
P 0 (2) = 97.8925
P 1 , d (2) = 99.5261
P 2 , d d (2) = 100
Table 9.4
Buy 0.8700 of bonds with maturity i = 2
Short 0.8554 of bonds with maturity i = 1
A Portfolio of bonds
=⇒
=⇒
Pay 0.8700 × $97.8925
Receive 0.8554 × $99.1338
=
=
$85.1703
$84.8007
Total Paid
=
$0.3697
where rK is the strike rate, e.g. rK = 2% and r1 denotes the interest rate at time 1. That is,
if the interest rate increases at i = 1, this security pays 100 × max (2% − 3.39%, 0) = $0,
while if the interest rate decreases at i = 1, this security pays 100×max (2% − 0.95%, 0) =
$1.05. That is,
(Payoff at i = 1 if r1,u ) = $0
(9.5)
(Payoff at i = 1 if r1,d ) = $1.05
What is the price of this security at time i = 0? What is the relation between this
security and the zero coupon bond in Table 9.3?
Consider the portfolio of bonds described in Table 9.4. In this portfolio, a trader is
long 0.8700of zero coupon bonds maturing on i = 2 and short 0.8554 zero coupon bonds
maturing on i = 1. These bond prices are in Table 9.1. The transaction implies a payment
of $85.1703 to buy the long-term bonds, and the receipt of $84.8007 from selling (short)
the 6-month bonds. The net payment is of $0.3697.
What is the value of the portfolio in Table 9.4 at time i = 1, in the two scenarios u and
d? Using the tree in Table 9.3 we find:
Value of Portfolio if r1,u
=
0.8700 × P1,u (2) − 0.8554 × $100 = $0
Value of Portfolio if r1,d
=
0.8700 × P1,d (2) − 0.8554 × $100 = $1.05
The values of the portfolio in the two scenarios are identical to the payoffs of the option
in Equations 9.5. That is, the portfolio that is long 0.8700 of the P0 (2) bond and short
340
ONE STEP BINOMIAL TREES
Table 9.5 The Option Value Tree
i=0
i=1
V1,u
=
=
100 × max (2% − 3.39%, 0)
$0
V1,d
=
=
100 × max (2% − 0.95%, 0)
$1.05
V0 = $0.3697
0.8554 of the P0 (1) bond replicates the payoff of the option. This finding implies that the
value of the option with payoffs in Equations 9.5 at time i = 0 must be exactly $0.3697,
that is, the cost of the portfolio obtained in Table 9.4.
Why must the option have the same value of the portfolio in Table 9.4? Otherwise, an
arbitrage opportunity arises: If for instance a trader is able to sell the option for $1, he or
she can then enter into an offsetting position by purchasing the portfolio in Table 9.4 for
$0.3697 and pocket today the difference of $0.6303. Because at time i = 1 the option is
perfectly hedged by the portfolio payoff, there is no risk, and today the trader has made a
profit. In well-functioning markets, such opportunities cannot be sustained. We therefore
obtain the tree for the option in Table 9.5.
This discussion prompts the following definition of a replicating portfolio:
Definition 9.1 A replicating portfolio of a security with payoffs V1,u and V1,d in the two
nodes u and d at time i = 1 is a portfolio of bonds that exactly replicates the values of the
security at time i = 1. That is, if Πi,j denotes the value of the portfolio at time i in node j,
we have Π1,u = V1,u and Π1,d = V1,d . The value of the option at i = 0 equals the value
of the portfolio Π0 = V0
9.2.1 The Replicating Portfolio Via No Arbitrage
The interesting fact about the portfolio in Table 9.4 is that it replicates exactly the payoff of
the security (option) whose payoffs are given in Equations 9.5. Is this a general property?
Can we always find a portfolio that replicates any payoff structure at time i = 1? We now
show that this is indeed a general property of binomial trees, which is the reason of their
popularity. Consider the generic tree of a security whose value at time i = 1 depends on
the interest rate, as given in Table 9.6.
Consider now a portfolio with N1 units of the bond with maturity i = 1 and N2 units
of the bond with maturity i = 2. We call this portfolio Πi,j at time i and node j. At time
i = 0, the value of the portfolio is then
Π0 = N1 × P0 (1) + N2 × P0 (2)
(9.6)
NO ARBITRAGE ON A BINOMIAL TREE
Table 9.6
341
The Tree of an Interest Rate Security
i=0
i=1
V1,u
V0
V1,d
where P0 (1) and P0 (2) are in Table 9.1. At time i = 1, the portfolio value will be
Π1,u
= N1 × 100 + N2 × P1,u (2)
Π1,d
= N1 × 100 + N2 × P1,d (2)
where the “100” is the value of P1 (1), and P1,u (2) and P1,d (2) are given in Table 9.3.
We want Π to replicate the value of the security V at time i = 1. Thus, we impose the
following two equations
Π1,u
= N1 × 100 + N2 × P1,u (2) = V1,u
(9.7)
Π1,d
= N1 × 100 + N2 × P1,d (2) = V1,d
(9.8)
This is a system of two equations in two unknowns (N1 and N2 ) which we can solve. In
particular, subtract Equation 9.8 from Equation 9.7, and factor N2 out to obtain
N2 × (P1,u (2) − P1,d (2)) = (V1,u − V1,d )
We can then solve for N2
V1,u − V1,d
P1,u (2) − P1,d (2)
Given N2 and Equation 9.7, we can solve for N1 to find
N2 =
(9.9)
1
× [V1,u − N2 × P1,u (2)]
(9.10)
100
Given N1 and N2 , the value of the portfolio at time i = 0 is given by Equation 9.6.
The next example shows that indeed the portfolio in Table 9.4 satisfies Equations 9.9
and 9.10, explaining why it exactly replicates the option’s payoffs in Equation 9.5.
N1 =
EXAMPLE 9.1
Consider the option payoff in Equation 9.5, which implies V1,u = $0 and V1,d =
$1.05. We can compute the replicating portfolio by using the binomial tree of the
2-period bond in Table 9.3. Applying the formulas in Equations 9.9 and 9.10 we
obtain:
0 − $1.05
V1,u − V1,d
=
= 0.8700
N2 =
P1,u (2) − P1,d (2)
$98.3193 − $99.5261
1
1
× (V1,u − N2 × P1,u (2)) =
× (0 − 0.8700 × 98.3193) = −0.8554,
N1 =
100
100
342
ONE STEP BINOMIAL TREES
which is the bond portfolio described in Table 9.4. The negative sign on N1 indicates
a short position in the bond with maturity i = 1.
The above methodology can be used to price any security that depends on the interest rate
r1 . The following examples illustrate the methodology.
EXAMPLE 9.2
Consider a swap that pays at time i = 1 the amount 100
2 × (r1 − c), where c is
the swap rate. Let c = 2%. Then, the value of the swap from the fixed rate payer
perspective is
V1,u
=
V1,d
=
100
× (3.39% − 2%) = $0.695;
2
100
× (0.95% − 2%) = −$0.525
2
We can obtain the replicating portfolio by choosing N1 and N2 according to Equations
9.10 and 9.9:
N2
=
N1
=
$0.695 − (−$0.525)
V1,u − V1,d
=
= −1.011
P1,u (2) − P1,d (2)
$98.3193 − $99.5261
1
1
× (V1,u − N2 × P1,u (2)) =
× (0.695 − (−1.011) × 98.3193) = 1.001
100
100
In this case, the replicating portfolio calls for a long position of 1.001 in the shortterm bond, and a short position of 1.011 in the long(er)-term bond. We can check
that the replicating portfolio in fact replicates:
Π1,u
= N1 × 100 + N2 × P1,u (2) = 1.001 × 100 − 1.011 × 98.3193 = $0.695
Π1,d
= N1 × 100 + N2 × P1,d (2) = 1.001 × 100 − 1.011 × 99.5261 = −$0.525
Because this portfolio indeed replicates the payoff from the swap, the value at i = 0
of the swap is
Π0 = N1 × P0 (1) + N2 × P0 (2) = 1.001 × 99.1338 − 1.011 × 97.8925 = $0.259
The interest rate dependence of a security can also stem from the value of a bond itself.
That is, a security that depends on the value of a bond implicitly also depends on the interest
rate. In this case, the no arbitrage relation still holds. Consider, for instance, the following
option on a bond.
EXAMPLE 9.3
Consider an option with a payoff at time i = 1 that depends on the zero coupon bond
that matures at time i = 2. In particular, let this option yield the following payoff at
time i = 1:
(Payoff at i = 1) = max (P1 (2) − K, 0)
NO ARBITRAGE ON A BINOMIAL TREE
343
where the strike price K = $99.00. In this case, the payoff at i = 1 from the option
is
V1,u
=
max(98.3193 − 99.00, 0) = 0;
V1,d
=
max(99.5261 − 99.00, 0) = $0.5261
Once again, we can obtain the replicating portfolio by choosing N1 and N2 according
to Equations 9.10 and 9.9, respectively:
N2
=
N1
=
0 − $0.5261
V1,u − V1,d
=
= 0.436
P1,u (2) − P1,d (2)
$98.3193 − $99.5261
1
1
× (V1,u − N2 × P1,u (2)) =
× (0 − 0.436 × 98.3193) = −0.429
100
100
In this case, the replicating portfolio calls for a short position of 0.429 in the shortterm bond, and a long position of 0.436 of the long(er)-term bond. Once again, we
can check that the replicating portfolio in fact replicates:
Π1,u
=
N1 × 100 + N2 × P1,u (2) = −0.429 × 100 + 0.436 × 98.3193 = $0
Π1,d
=
N1 × 100 + N2 × P1,d (2) = −0.429 × 100 + 0.436 × 99.5261 = $0.5261
Because this portfolio indeed replicates the payoff from the option, the value of the
option at i = 0 is
Π0 = N1 × P0 (1) + N2 × P0 (2) = −0.429 × 99.1338 + 0.436 × 97.8925 = $0.185
It is useful to summarize these results as follows:
Fact 9.1 On a binomial tree, any interest rate security with values V1,u and V1,d at time
i = 1 can be replicated by a (replicating) portfolio Π whose positions in bonds with
maturity i = 1 and i = 2 are given by N1 and N2 in Equations 9.10 and 9.9, respectively.
The price of the security at time 0 is then given by
V0 = N1 × P0 (1) + N2 × P0 (2)
(9.11)
We can then summarize the (first) recipe to price derivative securities:
Recipe 1:
1. Compute N1 and N2 from Equations 9.10 and 9.9.
2. Compute the price of the derivative security V0 according to Equation 9.11.
9.2.2
Where Is the Probability p?
The prices of the derivative securities discussed in previous pages were obtained without
any reference to the probability p of an up movement in interest rates (see Table 9.2). This
fact seems puzzling: How is possible that the price of an option that pays when interest
344
ONE STEP BINOMIAL TREES
rates go down, for instance, is independent of the probability that the rate will in fact go
down? The logic is the following: From Fact 9.1 the price of the derivative security is
computed from a portfolio of other bonds. The prices of these bonds do depend on the
probability that market participants assign to an increase in future interest rate. Everything
else equal, if market participants have a lower expectation of the 6-month rate next period,
then the long-term bond would have a higher price, which in turn would increase the price
of the option. Yet, for given bond prices, the price of the option can be computed only
by the replication of its payoff and thus the exact knowledge of the probability p is not
necessary.
9.3 DERIVATIVE PRICING AS PRESENT DISCOUNTED VALUES OF
FUTURE CASH FLOWS
The methodology developed in the previous section is relatively cumbersome. To obtain
the price of an interest rate security, we must first obtain the replicating portfolio, and then
obtain the price of the derivative security as the value of the portfolio. In Section 9.4 we will
develop a technique that substantially simplifies the computation, especially when we move
to longer (and more realistic) trees. However, before we get there, we need to introduce
another concept, namely, the pricing of derivatives as risk-adjusted discounted present
value of future cash flows. This intermediate step also clarifies much of the (sometimes
confusing) terminology adopted in the pricing of interest rate securities.
9.3.1 Risk Premia in Interest Rate Securities
The price of the zero coupon bond maturing at i = 2 is P0 (2) = 97.8925 (see Table 9.1).
Given the tree in Table 9.3, we can compute the present value of the expected bond price
at time i = 1, E[P1 (2)], using the risk free rate to discount to time i = 0. Let Δ = 1/2 be
the time interval between steps. Given p = 1/2, we have:
Present value of E[P1 (2)]
= e−r 0 ×Δ × E [P1 (2)]
(9.12)
=
0.9913 × (p × 98.3193 + (1 − p) × 99.5261)
=
98.0658
We find then that the price of the 2-period zero coupon bond is less than the present value
of its value at i = 1. That is, from Equation 9.12 we have
P0 (2) = 97.8925 < e−r 0 ×Δ × E [P1 (2)] = 98.0658
(9.13)
The price is lower because of a risk premium embedded in the price of longer term bonds.
What risk? Clearly, there is no default risk in U.S. Treasury bonds, as the U.S. government is extremely unlikely to default on its obligations. That is, an investment in Treasury
securities is safe if held to maturity. However, as discussed in Chapters 3, 4, and 7, an
investment in Treasury securities is risky because an investor would suffer capital losses if
the bond is sold before maturity after an increase in interest rates.
For later reference, let us define the (dollar) risk premium as follows:
DERIVATIVE PRICING AS PRESENT DISCOUNTED VALUES OF FUTURE CASH FLOWS
345
Definition 9.2 The dollar risk premium from investing in the long-term bond with maturity
i = 2 is defined as
Dollar risk premium = e−r 0 ×Δ × E [P1 (2)] − P0 (2)
(9.14)
The word “dollar” clarifies that this is a risk premium expressed in dollar units, rather
than in percentages. The dollar risk premium defined in Equation 9.14 is useful in binomial
trees, as shown below. In the example above we have:
Dollar risk premium = e−r 0 ×Δ × E [P1 (2)] − P0 (2) = 0.1733
9.3.2
(9.15)
The Market Price of Interest Rate Risk
We now establish a relation between derivative securities. First, consider again the derivation of N1 and N2 : recall that given N2 in Equation 9.9, we computed the value of N1
by substituting N2 into Equation 9.7, thereby obtaining Equation 9.10. Equivalently, we
could substitute the same value of N2 into Equation 9.8, in which case we would obtain
N1 = V1,d − N2 × P1,d (2). Clearly, the solution of the system of equation N1 is the same
independently of where we substitute. That is, we have
N1 × 100 = V1,u − N2 × P1,u (2) = V1,d − N2 × P1,d (2)
This implies that we can also write, equivalently
N1 =
1
× {E[V1 ] − N2 × E[P1 (2)]}
100
(9.16)
where E[V1 ] = pV1,u + (1 − p)V1,d and E[P1 (2)] = pP1,u (2) + (1 − p)P1,d (2). This
expression leads to an interesting relation. In fact, recall from Equation 9.11 that the value
of the security at time i = 0 is given by V0 = N1 × P0 (1) + N2 × P0 (2). Thus, we can
substitute N1 into Equation 9.16 to obtain
V0 = E[V1 ] ×
P0 (1)
P0 (1)
− N2 × E[P1 (2)] ×
+ N2 × P0 (2)
100
100
Factoring out N2 and rearranging, we obtain
P0 (1)
P0 (1)
− P0 (2) = E[V1 ] ×
− V0
N2 × E[P1 (2)] ×
100
100
Substitute N2 = (V1,u −V1,d )/(P1,u (2)−P1,d (2)) (see Equation 9.9), divide both sides by
(V1,u − V1,d ), and remember that P0 (1) = 100 × e−r 0 ×Δ , to obtain the following relation:
e−r 0 ×Δ E[V1 ] − V0
e−r 0 ×Δ E[P1 (2)] − P0 (2)
=
P1,u (2) − P1,d (2)
V1,u − V1,d
(9.17)
This is a key relation between securities in no arbitrage pricing. In fact, note the
following:
1. The left-hand side involves only the zero coupon bond prices, while the right-hand
side involves only the derivative security prices.
346
ONE STEP BINOMIAL TREES
2. The expression on the left-hand side for the 2-period bond is identical to the expression on the right-hand side for the derivative security.
3. The numerators of both expressions are nothing other than the (dollar) risk premium
investors require from holding bonds (on the left-hand side) or the derivative security
(on the right-hand side), as defined in Equation 9.14 for the 2-period bond.
4. The denominators represent instead the risk of an investment in bonds (on the lefthand side) or in the derivative security (on the right-hand side), as it is given by the
fluctuations of the security across the two possible states. For instance, the risk from
investing in the 2-period zero coupon bond is
Dollar Risk = P1,u − P1,d = −1.2068
(9.18)
Note that the negative sign is simply a reflection of the fact that the bond price
decreases when the interest rate increases. It is not wise to use absolute values here,
as for hedging purposes we need to know whether a security moves in the same
direction as or in the opposite direction as the interest rates. The negative sign shows
that it moves in the opposite direction.
Given this terminology, and the definition of a risk premium in Equation 9.14, Equation
9.17 says the following:
Fact 9.2 All interest rate securities on a binomial tree have the same ratio between risk
premium and risk. That is,
e−r 0 ×Δ E[V1 ] − V0
Risk premium
=
= λ0
Risk
V1,u − V1,d
(9.19)
where λ0 is common across all interest rate securities.
That is, whether we are considering a zero coupon bond with maturity k or a derivative
security, the ratio between risk premium and risk is always the same. This ratio has a name:
Definition 9.3 The ratio between risk premium and risk that is common across all interest
rate securities, λ0 in Equation 9.19, is called market price of (interest rate) risk.
9.3.3 An Interest Rate Security Pricing Formula
Fact 9.2 above has an important implication for the pricing of derivative securities. If we
know λ0 at time i = 0, we can compute the price of any security as
V0 = e−r 0 ×Δ × E[V1 ] − λ0 × (V1,u − V1,d )
(9.20)
How can we compute λ0 ? We can use the information about the bond with maturity i = 2
to compute it
e−r 0 ×Δ E[P1 (2)] − P0 (2)
(9.21)
λ0 =
P1,u (2) − P1,d (2)
This methodology leads to the second recipe to price derivative securities.
DERIVATIVE PRICING AS PRESENT DISCOUNTED VALUES OF FUTURE CASH FLOWS
347
Recipe 2:
1. Compute the market price of risk λ0 from Equation 9.21;
2. Compute the price of the interest rate security from the pricing formula in Equation
9.20.
For instance, from Table 9.3, we obtain:
λ0 =
98.0658 − 97.8925
e−r 0 ×Δ E[P1 (2)] − P0 (2)
=
= −0.1436
P1,u (2) − P1,d (2)
98.3193 − 99.5261
(9.22)
We can then apply this number to compute the price of any interest rate security by using
Equation 9.20. As an example, we can compute the value of the interest rate securities in
Examples 9.1 through 9.3as follows:
1. (Option)
(a) Present value of expected payoff = 0.9913 × (p × 0 + (1 − p) × 1.05) = 0.5205
(b) Risk adjustment = λ0 × (0 − 1.05) = 0.1508
(c) Value = (a) − (b) = 0.3697.
2. (Swap)
(a) Present value of expected payoff = 0.9913×[p×0.695+(1−p)×(−0.525)] =
0.084
(b) Risk adjustment = λ0 × [0.695 − (−0.525)] = −0.175
(c) Value = (a) − (b) = 0.259.
3. (Bond Option)
(a) Present value of expected payoff = 0.9913×[p×0+(1−p)×0.5261] = 0.261
(b) Risk adjustment = λ0 × [0 − 0.5261] = 0.076
(c) Value = (a) − (b) = 0.185.
The key is that once we know λ0 and the probability p, the pricing formula is always
identical, and it can be applied to price any interest rate derivative or security.
9.3.4
What If We Do Not Know p?
What if we make a mistake and miscalculate p? Does this imply that all of our calculations
go astray because of it? It seems like this methodology is inferior to the previous one based
on dynamic replication, as now we need to know p.
In fact, as it turns out, even if we make a mistake in computing p in the original tree,
the pricing of the interest rate securities is not affected. The key is to realize that λ0
also depends on p itself (see Equation 9.19). Thus, if we miscalculate p, we will also
miscalculate the risk adjustment. It turns out that one error exactly counterbalances the
other. For instance, Table 9.7 shows the value of the market price of risk λ0 , and the value
of the option discussed in Example 9.1 for values of the probability p ranging between 10%
348
ONE STEP BINOMIAL TREES
Table 9.7
The Value of the Option for Various Probabilities p
Probability
Present Value of
Expected Payoff
Market Price
of Risk
Risk
Adjustment
Value
at i = 0
p
e−r 0 ×Δ E[V1 ]
λ0
λ0 × (V1 , u − V1 , d )
V0
0.1
0.2
0.3
0.4
0.5
0.6
0.6448
0.7
0.8
0.9
0.9368
0.8327
0.7286
0.6245
0.5205
0.4164
0.3697
0.3123
0.2082
0.1041
-0.5401
-0.4410
-0.3419
-0.2427
-0.1436
-0.0445
0.0000
0.0547
0.1538
0.2529
0.5671
0.4630
0.3589
0.2549
0.1508
0.0467
0.0000
-0.0574
-0.1615
-0.2656
0.3697
0.3697
0.3697
0.3697
0.3697
0.3697
0.3697
0.3697
0.3697
0.3697
and 90%. The last column contains the key result: The value of the security is always equal
to $0.5385 independently of the probability p (in the first column).
Table 9.7 shows that as the probability p increases, the expected discounted value of
the future payoff decreases, as shown in Column 2 (recall that the option pays when the
interest rate declines). However, this effect is counterbalanced by another equally powerful
effect, and this is that the risk adjustment (Column 4) is also declining. The reason is
that an increase in p also changes the market price of risk λ0 , as shown in Column 3. To
understand this effect, recall that we compute λ0 from the price of the bond that expires at
time i = 2. In particular, an increase in the probability p decreases the expected discounted
value of the long-term bond e−r 0 ×Δ E[P1 (2)], and thus the risk premium in Equation 9.14.
Thus, the market price of risk λ0 becomes less negative as the probability p increases.
9.4 RISK NEUTRAL PRICING
Table 9.7 also shows that for p = 0.6448 the risk adjustment λ0 necessary to evaluate the
derivative security is zero. Denote this special probability p∗ . In this case, the pricing
Equation 9.20 is simply equal to
V0 = e−r 0 ×Δ × E ∗ [V1 ]
(9.23)
E ∗ [V1 ] = p∗ × V1,u + (1 − p∗ ) × V1,d
(9.24)
where
The pricing Equation 9.23 is much simpler to remember than Equation 9.20, as it is says
that the price of the option is simply equal to the discounted value of the future payoff,
in which we use the risk free rate to discount the payoff. In particular, the complicated
quantity λ0 , the market price of risk, is not there. However, recall, Equation 9.23 is
obtained by using a very special probability, p∗ = 0.6448, which is larger than the true
(original) probability p = 0.5, which underlies the original tree in Table 9.2. Yet, if we had
RISK NEUTRAL PRICING
349
a methodology to compute that particular probability p∗ which makes the pricing Equation
9.23 true, it would be a tremendous simplification for the pricing of interest rate securities.
9.4.1
Risk Neutral Probability
How can we compute this probability p∗ that makes Equation 9.23 true? Recall that this
probability is the one that makes the market price of risk λ0 = 0. Because λ0 is common
across all securities, it follows that the probability p∗ is the same for all securities. In
particular, the pricing Equation 9.23 must hold for the zero coupon bond
P0 (2)
= e−r 0 ×Δ × E ∗ [P1 ]
= e−r 0 ×Δ × (p∗ × P1,u (2) + (1 − p∗ ) × P1,d (2))
From the zero coupon bond tree in Table 9.3, we know P1,u and P1,d , while P0 (2) is known
from today’s bond price. Thus, we can solve for p∗ obtaining
p∗ =
er 0 ×Δ P0 (2) − P1,d (2)
P1,u (2) − P1,d (2)
(9.25)
Definition 9.4 The risk neutral probability p∗ is the particular value of the probability p
such that every interest rate security is given by the present value of future expected payoff,
where the present value is computed using the risk free rate. That is, such that Equations
9.23 and 9.24 hold. The risk neutral probability p∗ can be computed out of the current two
period bond tree in Table 9.3 from Equation 9.25.
9.4.2
The Price of Interest Rate Securities
The risk neutral probability p∗ and the risk neutral valuation formula allow us to obtain a
third (equivalent) methodology to obtain the price of an interest rate security.
Recipe 3:
1. Compute the risk neutral probability p∗ from Equation 9.25;
2. Compute the price of the interest rate security from the pricing formula in Equation
9.23.
For instance, the risk neutral probability p∗ in Table 9.7 is computed as follows
p∗ =
e0.0174/2 97.8925 − 99.5261
er 0 ×Δ P0 (2) − P1,d (2)
=
= 0.6448
P1,u (2) − P1,d (2)
98.3193 − 99.5261
(9.26)
We can then apply this risk neutral probability to compute the price of any interest rate
security by using Equation 9.23. As an example, the value of the interest rate securities in
Examples 9.1 through 9.1 can be computed as follows:
1. (Option) V0 = 0.9913 × [ p∗ × 0 + (1 − p∗ ) × 1.05 ] = 0.3697
2. (Swap) V0 = 0.9913 × [ p∗ × 0.695 + (1 − p∗ ) × (−0.525) ] = 0.259
3. (Bond Option) V0 = 0.9913 × [ p∗ × 0 + (1 − p∗ ) × 0.5261 ] = 0.185
It works beautifully.
350
ONE STEP BINOMIAL TREES
Table 9.8 The Swap Tree
i=0
V0sw ap
i=1
sw ap
V1,u
=
$0.695
sw ap
V1,d
=
−$0.525
= $0.259
9.4.3 Risk Neutral Pricing and Dynamic Replication
The simplicity of the risk neutral methodology is its main virtue. It is important to realize
that there is no underlying assumption that market participants are risk neutral. They
are not, as in fact under the true probability (p = 1/2 in the tree in Table 9.2), market
participants would require a risk premium to hold long-term bonds, given in Equation 9.14
in Section 9.3.1. The risk neutral methodology in Recipe 3 is just a convenient way of
using the no arbitrage argument to obtain the price of derivative securities. Underlying its
logic is the existence of the replicating portfolio in Definition 9.1.
What is key to realize, however, is that the dynamic replication strategy exists among
any two interest rate securities. For instance, we saw in Chapter 1 and 5, the swap market
has grown tremendously in the last two decades, and it is larger now, for instance, than the
U.S. Treasury bond market. In the context of our Example 9.2, we considered the swap
as a derivative security, whose price depends on the one of Treasuries. However, in our
model, if we happen to know the value of swaps, we can replicate the options, or even the
payoff of Treasuries, by using swaps instead. For instance, the next example shows the use
of swaps to replicate the option in Example 9.1. Risk neutral pricing here is instrumental
to derive the price of the swap. Given that, we can use it to replicate the option.
EXAMPLE 9.4
We are going to use a portfolio with N2 swaps (as in Example 9.2), and N1 of 1-period
bonds. The methodology is the same as in Section 9.2.1, namely, Equations 9.9 and
9.10, with the only difference that instead of the value of bonds, we must use the
value of the swap along the way. In particular, let Vijsw ap be the value of the swap in
node ij. In Example 9.2 we obtained V0 = $0.259, and thus the swap binomial tree
is as in Table 9.8.
Using the formulas for N1 and N2 in Equations 9.10 and 9.9, respectively, we
obtain that the portfolio of a swap and short-term bond is given by
N2
=
N1
=
− V1,d
0 − 1.05
= −0.861
sw ap =
− V1,d
0.695 − (−0.525)
#
1
1
sw ap $
× V1,u − N2 × V1,u
× [0 − (−0.861) × 0.695] = 0.006
=
100
100
V1,u
sw ap
V1,u
RISK NEUTRAL PRICING
351
The value of this portfolio is
Π0 = N1 × P0 (1) + N2 × V0sw ap = 0.006 × $99.1338 − 0.861 × $0.259 = $0.3697
The value of the portfolio is identical to the value of the option that was obtained by
using the portfolio of bonds, as in Table 9.4. Does this replicating portfolio replicate?
Π1,u
sw ap
= N1 × P1,u (1) + N2 × V1,u
= 0.006 × 100 − 0.861 × $0.695 = $0
Π1,d
sw ap
= N1 × P1,d (1) + N2 × V1,d
= 0.006 × 100 − 0.861 × $(−0.525) = $1.05
It replicates the option in Table 9.5 exactly.
9.4.4
Risk Neutral Expectation of Future Interest Rates
The expected future interest rate under the risk neutral probability is given by
E ∗ [r1 ] = p∗ ×r1,u +(1−p∗ )×r1,d = 0.6448×3.39%+0.3552×0.95 = 2.5234% (9.27)
This number is much higher than the true expected interest rate computed in Equation
9.1, which was equal to E[r1 ] = 2.17%. Risk neutral pricing is tantamount to including
the risk premium in the probability of an up move (from p to p∗ ) and thus increasing the
predicted future interest rates. However, this does not mean that market participants expect
the interest rate in six months to be 2.5234%, instead of 2.17%. The higher risk neutral
predicted rate is due only to the higher discount that market participants require to hold
risky securities.
We are going to see that to some extent, the risk neutral expectation of future interest
rates is close to the forward rate. Indeed, from the data in Table 9.1, the continuously
compounded forward rate between period i = 1 and i = 2 is given by
P0 (1)
1
= 2.52%
f (0, 1, 2) = − × ln
2
P0 (2)
This is indeed very close to the risk neutral expected future interest rate in Equation 9.27.
Note, in particular, that the forward rate is quite different from the true expected future
interest rate E[r1 ] = 2.17%, as discussed in Chapter 7 (see e.g. Example 7.2). This finding
has two implications:
1. Forward rates are not equal to the market expectation of future interest rates. In this
model, the former is 2.52% while the latter is only 2.17%.
2. The forward rate (=2.52%) is not even equal to the risk neutral expected future
interest rate (=2.5234%), although they are quite close.
Point 1 implies that we should be careful in interpreting futures rates as expected future
interest rates, because although there is a relation, they are not the same. Thus, if today
we observe high forward rates we should think about two possibilities: Either market
participants expect higher future interest rates; or they are strongly averse to risk, and thus
the price of long term bonds is low today. From our earlier discussion, high aversion to
risk manifests itself in a high (negative) market price of risk λ0 or, equivalently, on a high
352
ONE STEP BINOMIAL TREES
risk neutral probability p∗ . The evidence presented in Chapter 7 shows that the second
interpretation is more consistent with the data.
Point 2 states that the forward rate is close to the risk neutral expected future interest
rate. Although the equality is much closer than with true probabilities, the two numbers are
not exactly the same. Briefly, the reason is that risk neutral pricing is based on the notion of
dynamic replication, as discussed in Section 9.2.1, which involves the trading of securities.
Thus, the important quantities are the prices at which transactions occur. Interest rates,
as discussed many times, only represent prices through a convex relation. The fact that
there is not a linear relation between prices and interest rates is the source of the difference
between expected future interest rates E ∗ [r1 ] and forward rates f (0, 1, 2). This can be seen
directly from the risk neutral pricing formula (9.23), which implies that for zero coupon
bonds
P0 (2) = e−r 0 ×Δ E ∗ [P1 (2)]
From the definition of forward rate, however, we also have
P0 (2) = e−r 0 ×Δ e−f (0,1,2)×Δ × 100
Comparing these two equations, because both must be true, and because we can write
E ∗ [P1 (2)] = E ∗ [e−r 1 ×Δ ] × 100 it follows that
e−f (0,1,2)×Δ = E ∗ [e−r 1 ×Δ ]
(9.28)
Finally, note that the exponential function e−r 1 ×Δ is a decreasing convex function of the
rate r1 . Thus, Jensen’s Inequality implies that3
E ∗ [e−r 1 ×Δ ] > e−E
∗
[r 1 ]×Δ
(9.29)
Equality 9.28 and Inequality 9.29 imply that
e−f (0,1,2)×Δ > e−E
∗
[r 1 ]×Δ
which implies
f (0, 1, 2) < E ∗ [r1 ]
(9.30)
Note that this is indeed true in the above example, in which f (0, 1, 2) = 2.52% and
E ∗ [r1 ] = 2.5234%. This discussion also suggests an adjustment to be made to the risk
neutral expected future interest rate to make it closer to the forward rate, and we will discuss
this adjustment – a popular practice in the industry – in Chapter 21.
9.5 SUMMARY
In this chapter we covered the following topics:
1. One step interest rate binomial tree: This type of binomial tree describes the two
possible scenario for the short term-rate next period.
3 Jensen’s
inequality states that if f (x) is a convex function of a variable x, then E[f (x)] > f [E(x)].
EXERCISES
353
2. Bond price binomial tree: From the two scenarios for one-period interest rates next
period, we can compute the two scenarios of the one-period bond next period. Using
today’s current quoted price of this bond, we obtain the tree for a two-period bond.
3. Replicating portfolio: This is a portfolio made up of a long-term security and shortterm bond that replicates the payoffs of another interest rate security in the two
possible interest rate scenarios next period. For instance, a long-term bond and a
short-term bond in appropriate proportion can replicate the payoff of an interest rate
option. The value of the replicating portfolio must equal the value of the derivative
security, otherwise an arbitrage opportunity arises.
4. Market price of risk: No arbitrage requires that all long-term securities must have
the same ratio of expected return to risk, otherwise an arbitrage opportunity arises.
5. Risk neutral probability: This is a “fake” probability that enables us to obtain a
convenient formula for the pricing of derivative securities by no arbitrage. Given
this “fake” probability p∗ , the value of any interest rate security on the binomial tree
equals the expected present value of the future payoff discounted at the risk free rate.
p∗ differs from the real probability of an upward movement in the interest rate as it
contains a component that is due to the risk premium that investors require to hold
long-term bonds.
9.6
EXERCISES
1. Consider the interest rate tree in Table 9.9.
(a) Compute the expected 6-month Treasury rate E[r1 ].
(b) The 1-year Treasury bill is trading at P0 (2) = 97.4845. What is the (continuously compounded) forward rate for the periods i = 1 to i = 2? How does it
compare with the expected rate computed in Part (a)? Explain.
(c) Compute the market price of risk λ. Interpret.
(d) Compute the risk neutral probability p∗ . Interpret.
2. Consider again Exercise 1 and the interest rate tree in Table 9.9.
(a) Consider an option with payoff
Option payoff at 1 = 100 × max(r1 − 2%, 0)
Compute the value at time i = 0 of the option by using the three methodologies
discussed in Sections 9.2.1, 9.3.3 and 9.4.2.
(b) Use the risk neutral pricing methodology to compute the value of a bond option
with payoff
Bond option payoff at 1 = max(P1 (2) − 98.5, 0)
3. Consider the tree in Table 9.10. You estimated the risk neutral probability to move
up the tree to be p∗ = 1/2.
354
ONE STEP BINOMIAL TREES
Table 9.9 A One-Step Binomial Interest Rate Tree
period =⇒
time (in years) =⇒
i=0
t=0
i=1
t = 0.5
r1,u = 4%
with prob. p = 1/2
r1,d = 1%
with prob. 1 − p = 1/2
r0 = 2%
Table 9.10
period =⇒
time (in years) =⇒
i=0
t=0
A One-Period Risk Neutral Interest Rate Tree
i=1
t = 0.5
r1 , u = 6%
with risk neutral prob. p ∗ = 1/2
r1 , d = 3%
with risk neutral prob. 1 − p ∗ = 1/2
r0 = 4%
(a) Compute the value of the zero coupon bonds maturing at time i = 1 and at
i = 2.
(b) Compute the continuously compounded yields for both bonds.
(c) Compute the value of an option with payoff
Option Payoff at 1 = 100 × max(r1 − 4%, 0)
(d) Set up the replicating portfolio that uses the bond prices determined in Part (a),
that is able to replicate the option’s payoff. Check that this portfolio in fact
replicates the option.
(e) Given the tree for the option, set up a replicating portfolio made of the shortterm bond and the option that is able to replicate the prices of the long-term
bond at time 1, that is, P1,u (2) and P1,d (2).
4. In the previous exercise, do you have enough information to compute the market
price of risk λ, and therefore the expected return on a bond maturing at time i = 2?
Explain.
5. The current 6-month and 1-year Treasury bills are trading at Pbill (0, 0.5) = 97.531
and Pbill (0, 1) = 95.1241, respectively. Consider now a binomial tree with root
r0 , and r1,u and r1,d as two interest rate scenarios after an upward or downward
movement in the interest rate, respectively.
EXERCISES
355
(a) What is r0 ?
(b) Let p∗ = 0.5 be the risk neutral probability. Do you have enough information
to pin down a unique value of r1,u and r1,d ? Provide at least three examples
of values of the pair (r1,u , r1,d ) that are consistent with the two bond prices
above. What is the difference across the three pairs of values?
(c) An option with payoff
Option payoff at 1 = 100 × max(r1 − 5%, 0)
is also trading at C0 (1) = $0.97531. Do you now have sufficient information
to pin down the pair (r1,u , r1,d )? Explain.
(d) Suppose you did not know p∗ = 0.5. What other information would you need
to compute also p∗ ? Provide an example.
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CHAPTER 10
MULTI-STEP BINOMIAL TREES
We now move to multi-step binomial trees. The key is to realize that a multi-step tree is
nothing more than a sequence of one-step trees. Therefore, any argument we made for each
little tree in Chapter 9 holds here too.
10.1
A TWO-STEP BINOMIAL TREE
Consider extending the binomial tree in Table 9.2 in Chapter 9 to one more period. Recall
that the interest rates on the tree are continuously compounded (see discussion in Section
9.1.1 in Chapter 9). The binomial tree in Table 10.1 is recombining, which means that
an “up and down” movement in interest rates leads to the same level as a “down and up”
movement. This, of course, need not be the case in general. But using recombining trees
becomes particularly helpful when we move to very long trees, with hundreds of steps.
Non-recombining trees require massive computing power to be solved.
Finally, we assume that the probability p of an up movement is constant and equal to 1/2
along the tree. This assumption is not necessary, and it is made here only for convenience.
This interest rate tree was computed on January 8, 2002, in a way that its implied forecasts
of future interest rates are reasonable given the information available at that time. In
particular, since the top node uu is reached with probability p × p = 1/4, the bottom one
dd is reached with probability (1 − p) × (1 − p) = 1/4, and the middle with probability
357
358
MULTI-STEP BINOMIAL TREES
Table 10.1
period =⇒ i = 0
time =⇒
t=0
A Two Step Interest Rate Tree
i=1
t = 0.5
i=2
t=1
r2,u u = 5.00%
r1,u = 3.39%
r2,u d
= 2.56%
r2,du
r0 = 1.74%
r1,d = 0.95%
r2,dd = 0.11%
probability of “up” movement: p = 1/2
Table 10.2
Treasury STRIPS on January 8, 2002
Maturity
Price
Yield
0.5
1
1.5
99.1338
97.8925
96.1462
1.74 %
2.13 %
2.62 %
Source: The Wall Street Journal.
2 × p × (1 − p) = 1/2, we have the predicted rate in six and twelve months
E [r1 ]
=
E [r2 ]
=
1
1
r1,u + r1,d = 2.17%
2
2
1
1
1
r2,u u + r2,u d + r2,dd = 2.6%
4
2
4
Given the current 6-month interest rate of 1.74%, much lower than the post-war average
rate of about 5%, it was reasonable to expect a substantial increase in the 6-month interest
rate within the following year, all the way up to 2.56%.
The zero coupon term structure of interest rates on January 8, 2002, is in Figure 9.1 in
Chapter 9, while the STRIPS up to maturity T = 1.5 are reported in Table 10.2.
10.2
RISK NEUTRAL PRICING
The two key results discovered in Chapter 9 were:
RISK NEUTRAL PRICING
359
1. We can obtain the price of any interest rate security at time 0 by using the risk neutral
approach, that is
V0 = e−r 0 ×Δ × [p∗ × V1,u + (1 − p∗ ) × V1,d ]
(10.1)
2. We can replicate the payoff of any interest rate security by using another interest rate
security.
We now apply these concepts to binomial trees with more steps (two, for the moment).
In particular, we found in Chapter 9 that given the interest rate tree in Table 10.1 and the
zero coupon bonds in Table 10.2 the risk neutral probability of moving up at time i = 0 is
p∗ = 0.6448. Assume that like the true probability p in the interest rate tree in Table 10.1,
the risk neutral probability p∗ is constant as well on the tree. What is the price, say, of a
zero coupon bond maturing on i = 3 under these conditions? The next section answers
these questions.
10.2.1
Risk Neutral Pricing by Backward Induction
We can compute the price of the 3-period bond by starting from the end of the tree, and
applying the one-step risk neutral pricing formula repeatedly. Recall the notation first: Let
Δ = 0.5 denote the time step and
Pi,j (k) = Bond price at time i in node j with maturity k
From the interest rate tree in Table 10.1 the bond prices in the three nodes at time i = 2 are
given by
P2,u u (3) = e−r 2 , u u ×Δ × 100 = 97.5310
P2,u d (3)
= e−r 2 , d u ×Δ × 100 = 98.7282
P2,du (3)
P2,dd (3)
= e−r 2 , d d ×Δ × 100 = 99.9450
What about the prices at time i = 1? Moving backward on the tree, we can apply the
formula in Equation 10.1, but rather than time i = 0, we use time i = 1, in both nodes
(1, u) and (1, d). In this case, the formula for bonds in the two nodes becomes
(1, u) node: P1,u (3)
= e−r 1 , u ×Δ × [p∗ × P2,u u (3) + (1 − p∗ ) × P2,u d (3)] (10.2)
= 0.9831 × [0.6448 × 97.5310 + 0.3552 × 98.7282]
= 96.3098
(1, d) node: P1,d (3)
= e−r 1 , d ×Δ × [p∗ × P2,du (3) + (1 − p∗ ) × P2,dd (3)] (10.3)
= 0.9951 × [0.6448 × 98.7282 + 0.3552 × 99.9450]
= 98.6904
We can now use these two prices to compute the price today
P0 (3)
= e−r 0 ×Δ × [p∗ × P1,u (3) + (1 − p∗ ) × P1,d (3)]
= 0.9913 × [0.6448 × 96.3098 + 0.3552 × 98.6904]
= 96.3137
(10.4)
360
MULTI-STEP BINOMIAL TREES
Table 10.3
i=0
t=0
The 3-Period Zero Coupon Bond Tree with Constant p∗ = 0.6448
i=1
t = 0.5
i=2
t=1
i=3
t = 1.5
P 3 , u u u (3) = 100
P 2 , u u (3) = 97.5310
)
P 3 , u u d (3)
P 3 , u d u (3)
P 3 , d u u (3)
P 1 , u (3) = 96.3098
*
P 2 , u d (3)
P 2 , d u (3)
P 0 (3) = 96.3137
= 100
= 98.7282
)
P 3 , u d d (3)
P 3 , d u d (3)
P 3 , d d u (3)
P 1 , d (3) = 98.6904
= 100
P 2 , d d (3) = 99.9450
P 3 , d d d (3) = 100
The price of the bond expiring at i = 3 is P0 (3) = $96.3137. Table 10.3 plots the resulting
zero coupon bond tree.
The risk neutral pricing methodology is very convenient, as it provides a simple rule to
obtain prices of interest rate securities. However, recall the two key ingredients to use the
risk neutral pricing techniques recipe:
1. The interest rate tree (e.g., Table 10.1).
2. The risk neutral probability (e.g. p∗ = 0.6448)
Given these two quantities, we can compute the price of any interest rate security. Let’s
do this by looking at the following example.
EXAMPLE 10.1
Consider a security that pays at i = 2, (t = 1), the following amount
Payoff at i = 2 : V2 = max (P2 (3) − K, 0) + max (K − P2 (3), 0)
where K = 98.7282. This payoff is given by a combination of a long call and a long
put with the same strike price, an investment strategy called a Straddle. A straddle
pays little when the bond price at maturity is close to the common strike price K,
361
RISK NEUTRAL PRICING
Table 10.4
period =⇒ i = 0
time =⇒
t=0
A Straddle
i=1
t = 0.5
i=2
t=1
V2,u u = 1.1972
V1,u = 0.7590
V2,u d
=0
V2,du
V0 = 0.6366
V1,d = 0.4301
V2,dd = 1.2169
but it pays handsomely if the price is far away from the strike, independently of
whether it is high or low. Because it pays when the price is far away from K, this
strategy tends to payoff in an environment with high bond price volatility. The risk
neutral pricing methodology can provide the price of this security immediately by
using a backward methodology. Namely, we start from the end of the tree (i = 2)
and compute the payoffs of the security. And then we move backward on the tree by
adopting Equation 10.1 in the various nodes. The risk neutral probability p∗ to use
is the same as above, namely, p∗ = 0.6448, as this probability applies to all interest
rate securities. Table 10.4 contains the tree in this case. The price is V0 = 0.6366.
10.2.2
Dynamic Replication
It is important to recall that behind the risk neutral pricing approach there is an underlying
replication strategy that allows a trader to replicate the final payoff by a portfolio of longterm and short-term bonds. In multistep trees, this portfolio needs to be rebalanced over
time, as interest rate change. In this sense, the strategy is called dynamic replication
strategy, or dynamic hedging strategy, as the strategy now involves the sale and purchases
of bonds over time, as we move along the tree.
More specifically, the dynamic replication strategy involves positions in long-term and
L
be
short-term bonds that change along the binomial tree. At each node (i, j), we let Ni,j
S
the position in the bond expiring at i = 3, and Ni,j be the position in the one-period bond
expiring at i + 1.
L
S
and Ni,j
? The key is that a long tree tree is a sequence of little
How do we choose Ni,j
one-step binomial trees. Thus, the formulas are the same as (9.9) and (9.10) in Chapter 9,
but applied along the longer tree. For instance, if we are at time i = 1 after an up movement,
i.e., (i, j) = (1, u), then the replicating portfolio calls for the following positions:
L
N1,u
=
V2,u u − V2,u d
P2,u u (3) − P2,u d (3)
(10.5)
362
MULTI-STEP BINOMIAL TREES
S
N1,u
=
$
1 #
L
V2,u u − N1,u
× P2,u u (3)
100
(10.6)
and similarly in the node (1, d).
Consider Example 10.1, for instance. The following trading strategy replicates the
Straddle in Table 10.4.
EXAMPLE 10.2
Time i = 0: At time zero, the dynamic replication calls for
0.7590 − 0.4301
V1,u − V1,d
=
P1,u (3) − P1,d (3)
96.3098 − 98.6904
= −0.1381
$
1 #
1
V1,u − N0L × P1,u (3) =
[0.7590 − (−0.1381) × 96.3098]
=
100
100
= 0.1406
=
N0L
N0S
The value of the replicating portfolio is
Π0 = N0S ×P0 (1)+N0L ×P0 (3) = 0.1406×99.1338−0.1381×96.3137 = 0.6366
Of course, the value of the portfolio is the same as the one we obtained from risk
neutral pricing. Just to check, what happens to the portfolio at time i = 1 in the two
nodes, u and d?
Π1,u
= N0S × 100 + N0L × P1,u (3) = 0.1406 × 100 − 0.1381 × 96.3098
=
Π1,d
0.7590 = Vi,u
= N0S × 100 + N0L × P1,d (3) = 0.1406 × 100 − 0.1381 × 98.6904
=
0.4301 = Vi,d
The portfolio, so far, replicates the straddle in Table 10.4. What happens next?
Time i = 1: At time i = 1, the new position in long-term bonds depends on whether
the interest rate went up or down. Consider the two cases:
Node (1, u): Apply Equations 10.5 and 10.6 again, to obtain
L
S
= −1; N1,u
= 0.9873
N1,u
S
where note that N1,u
is the position in bond P1,u (2), expiring at time i = 2,
and whose value is P1,u (2) = 98.3193 (see Table 9.3 in Chapter 9). How much
ew
:
does this portfolio cost? Let’s denote this portfolio Πn1,u
ew
Πn1,u
=
S
L
N1,u
× P1,u (2) + N1,u
× P1,u (3)
=
0.9873 × 98.3193 − 1 × 96.3098
=
0.7590
RISK NEUTRAL PRICING
363
The value of the portfolio equals the value of the security we are replicating. In
particular, this portfolio value is also identical to the amount delivered in this
node (1, u) by the replicating portfolio formed at time 0. That is, we have
ew
Π1,u = Πn1,u
This implies that the old portfolio will deliver exactly enough money to buy the
new portfolio. In other words, the trading strategy is self financing.
ew
replicate the final payoff at time i = 2?
Does portfolio Πn1,u
S
L
= N1,u
× 100 + N1,u
× P2,u u (3) = 0.9873 × 100 − 1 × 97.5310
ew
Πn2,u
u
=
1.1972
S
L
= N1,u
× 100 + N1,u
× P2,u d (3) = 0.9873 × 100 − 1 × 98.7282
ew
Πn2,u
d
=
0
Yes, it works.
Node (1, d): Apply Equations 10.5 and 10.6 but to the (1, d) node, to obtain:
L
S
= 1; N1,u
= −0.9873
N1,u
The cost of the new portfolio is now (recall from Table 9.3 in Chapter 9 that
P1,d (2) = 99.5261):
ew
Πn1,d
=
S
L
N1,d
× P1,d (2) + N1,d
× P1,d (3)
= −0.9873 × 99.5261 + 1 × 98.6904
=
0.4301
ew
, and thus that the old portfolio
This implies, as before, that Π1,d = Πn1,d
delivers enough cash to form the new portfolio. At time i = 2, then
ew
Πn2,du
S
L
= N1,d
× 100 + N1,d
× P2,du (3) = −0.9873 × 100 + 1 × 98.7282
=
ew
Πn2,dd
0
S
L
= N1,d
× 100 + N1,d
× P2,dd (3) = −0.9873 × 100 + 1 × 99.9450
=
1.2169
It works.
Table 10.5 shows the dynamic portfolio strategy over time. Although this procedure
appears cumbersome, computers can be programmed to carry out the replication
strategy automatically. Indeed, program trading has become a standard tool for
proprietary trading desks and hedge funds. As trading strategies become increasingly
complex, more and more powerful computers are employed to help decide the optimal
trading strategy in order to achieve some goal, such as the replication of the straddle
in Example 10.1.
364
MULTI-STEP BINOMIAL TREES
Table 10.5 Dynamic Replication
i=0
t=0
i=1
t = 0.5
i=2
t=1
− − − − − − −Rebalance − − − − − −−
L
N1,u
= −1
S
N1,u
= 0.9873
ew
Πn2,u
u = 1.1972
N0L = −0.1381
N0S = 0.1406
Π1,u = 0.7590
L
−→ N1,u
= −1
S
−→ N1,u
= 0.9873
ew
−→ Πn1,u
= 0.7590
L
N1,u
= −1
S
= 0.9873
N1,u
ew
Πn2,u
d =0
L
−→ N1,u
=1
S
−→ N1,u
= −0.9873
ew
−→ Πn1,d
= 0.4301
L
N1,d
=1
S
N1,d
= −0.9873
n ew
Π2,du = 0
N0L = −0.1381
N0S = 0.1406
Π0 = 0.6366
N0L = −0.1381
N0S = 0.1406
Π1,d = 0.4301
L
N1,d
=1
S
N1,d
= −0.9873
ew
Πn2,dd
= 1.2169
MATCHING THE TERM STRUCTURE
10.3
365
MATCHING THE TERM STRUCTURE
The binomial tree in Table 10.3 was meant to illustrate the risk neutral methodology in
longer trees. To this end, we assumed that the risk neutral probability p∗ was the same
along the tree. However, there is a problem, namely, that the price of the zero coupon
bond that the tree produces, P0 (3) = 96.3137, is different from the traded prices on the
same day (January 8, 2002). In fact, from Table 10.2, the price of the bond at that time
was P0 (3) = 96.1462, which is lower. Since the risk neutral probability p∗ = 0.6448
was computed from the 2-period bond P0 (2), there is no a priori reason why it should be
constant along the tree itself.
Assume then that p∗ may change over time and denote by p∗i its value at time i. The
risk neutral probability obtained earlier is then denoted by p∗0 = 0.6448. The risk neutral
probability at time i = 1 is denoted p∗1 . In the example above we have then simply assumed
p∗1 = p∗0 , but this restriction is unnecessary.
How can we then compute p∗1 ?
Unfortunately, it is not possible to find a simple formula like the one developed in
Chapter 9 (see Equation 9.25). Instead, we must search for the value p∗1 that yields the
correct price of the 3-period bond. Table 10.6 shows the zero coupon bond value obtained
from the tree for various values of p∗1 , ranging from 0.1 to 0.9. More specifically, for
each probability p∗1 in the first column, the computation of the 3-period bond is performed
exactly as in Section 10.2.1, but with the only difference that when we compute P1,u (3)
and P1,d (3), we use p∗1 in Equations 10.2 and 10.3 instead of p∗ . The second column in
Table 10.6 shows that the price of the 3-period zero coupon bond decreases as p∗1 increases,
ranging from 96.9560 for p∗1 = .1 to 96.0129 for p∗1 = .9.
Table 10.6 also shows that when p∗1 = 0.7869, the model price of the zero coupon bond
is 96.1462, which is equal to the current traded price in Table 10.2. The corresponding
binomial tree for the 3-period bond is in Table 10.7.
Along with the risk neutral probability p∗1 , the value of derivative securities (and the
dynamic replication) also change. For instance, the value of the security described in
Example 10.1 is now given in Table 10.8.
10.4
MULTI-STEP TREES
We now extend the model to trees with longer maturity, or with more frequent steps. To
build multi-steps trees, we need a methodology. The following procedure is particularly
convenient.
1. Define the predicted future interest rate E[ri ] for many future horizons i = 1, 2, ..., n.
2. Define some errors of the predictions (e.g., r1,u = 3.39% and r1,d = 0.95% are
errors around the expected rate E[r1 ] = 2.17%).
3. Find the risk neutral probabilities that price bonds.
366
MULTI-STEP BINOMIAL TREES
Table 10.6
The 3-Period Zero Coupon Bond Price for Risk Neutral Probabilities p∗1
Table 10.7
i= 0
t= 0
RN Probability
p∗1
Model Price
P0 (3)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.7869
0.8
0.9
96.9560
96.8381
96.7202
96.6024
96.4845
96.3666
96.2487
96.1462
96.1308
96.0129
The 3-Period Zero Coupon Bond Tree that Matches the Bond Data
i= 1
t = 0.5
i= 2
t= 1
i= 3
t = 1.5
P 3 , u u u (3) = 100
P 2 , u u (3) = 97.5310
P 1 , u (3) = 96.1426
*
P 2 , u d (3)
P 2 , d u (3)
P 0 (3) = 96.1462
P 3 , u u d (3)
P 3 , u d u (3)
P 3 , d u u (3)
= 100
= 98.7282
P 3 , u d d (3)
P 3 , d u d (3)
P 3 , d d u (3)
P 1 , d (3) = 98.5184
= 100
P 2 , d d (3) = 99.9450
P 3 , d d d (3) = 100
367
MULTI-STEP TREES
Table 10.8
The Price of the Straddle with p∗1 = 0.7896
period =⇒ i = 0
time =⇒
t=0
i=1
t = 0.5
i=2
t=1
V2,u u = 1.1972
V1,u = 0.9263
V2,u d
=0
V2,du
V0 = 0.6830
V1,d = 0.2580
V2,dd = 1.2169
Step 1 depends on our best predictions of future monetary policy actions, and therefore,
our predictions of future inflation, real GDP growth, and so on. These predictions may be
based on statistical models, or even simply our forecasts of future rates. As in Chapter 7
we use data on 6-month rates from December 1961 to December 2001 and run a regression
on rt+ 1 onto rt . Given the estimated parameters, and the current interest rate rJ an 2002 =
1.74%, we can forecast the future interest rate as in Table 10.9. Specifically, the third
column of this table reports the forecasts of the 6-month rate from July 2002 to January
2007. The prediction, according to the regression model, is that the interest rate will rise
steadily in the next five years. Of course, predictions may be wrong, and they would have
been in 2002: the Federal Reserve, concerned with the weakening of the U.S. economy,
decreased the Federal funds rate further in 2002 and 2003. As a consequence, the realized
(ex post) 6-month rate also decreased, from 1.71% in July 2002 to 0.95% in July 2003.
However, rates climbed up quite rapidly after that, shooting up to 4.35% by January 2006,
almost matching the model forecast (of 4.22% in the third column). For July, 2006 to
January, 2007, the model’s forecasts were even lower than the realized rate.
10.4.1
Building a Binomial Tree from Expected Future Rates
We now take these expected rates and build a binomial tree around them. Define the
expected change in interest rate in the future as
mi = E[ri+ 1 − ri ]
(10.7)
We then introduce errors in our predictions as follows:
r1,u
r1,d
√
= r0 + m0 × Δ + σ Δ
√
= r0 + m0 × Δ − σ Δ
(10.8)
(10.9)
368
MULTI-STEP BINOMIAL TREES
Table 10.9
Interest Rate Prediction as of 8 January 2002
Semester
i
Month/Year
Model
Prediction
Realized Rate
Ex Post
1
2
3
4
5
6
7
8
9
10
Jul 2002
Jan 2003
Jul 2003
Jan 2004
Jul 2004
Jan 2005
Jul 2005
Jan 2006
Jul 2006
Jan 2007
2.17 %
2.56 %
2.91 %
3.22 %
3.51 %
3.77 %
4.00 %
4.22 %
4.41 %
4.58 %
1.71 %
1.20 %
0.95 %
0.97 %
1.67 %
2.62 %
3.45 %
4.35 %
5.12 %
5.01 %
Data Source: Federal Reserve.
for the first period, where the up or down movements occur with probability 1/2. The
second period, similarly,
√
r2,u u = r1,u + m1 × Δ + σ Δ
√
r2,u d = r1,u + m1 × Δ − σ Δ
√
r2,du = r1,d + m1 × Δ + σ Δ
√
r2,dd = r1,d + m1 × Δ − σ Δ
and so on. This model naturally generates a recombining tree. In fact, note that by
substituting r1,u and r1,d from the first equations, we find
r2,u d = r0 + (m0 + m1 ) × Δ = r2,du .
(10.10)
That is, an up and down movement leads to the same rate as a down and up movement.
From the construction, it also follows that r2,u d = r2,du = E[r2 ], i.e. 2.56% from Table
10.9.
Table 10.10 reports the resulting interest rate tree, under the assumption that σ = 0.0173,
as estimated from the same data from 1961 to 2001. To interpret the entries in this table,
we need to clarify the convention we are going to denote the points on the tree. That is, the
earlier notation u, uu, and so on is fine for small trees. But when we move to long trees, we
must use a slightly different notation. The tree in Table 10.10 shows that each point on the
tree can be described by a time index i and a node index j. Looking at the tree, an upward
movement is described by an increase in the index i, but not in the index j. A downward
movement is described by both an increase in the index i and j. That is:
ri+ 1,j
an upward movement in interest rate
(10.11)
ri,j −→
ri+ 1,j + 1 a downward movement in interest rate
In particular, it is important to remember that from any point in the tree, there are only
two possibilities: either go across, or go down. For instance, from the interest rate 2.56
MULTI-STEP TREES
369
Table 10.10 Interest Rate Tree
i =⇒
j
0
0
1
2
3
4
5
1.74 →
3.39 →
5.00 →
6.58 →
8.12 →
→
0.11 1.68 →
3.22 →
→ 11.11 → 12.57 → 14.00
9.63 →
→
7.18 →
8.66
10.12
11.56
4.73 →
6.22 →
7.67 →
9.11
1
0.95 →
2.56 →
2
3
4.13 →
→
−0.76 4
5
6
7
8
5.67 →
→
0.78 6
→
2.29 →
3.77 7
→
5.23 8
6.66
→ −0.16 → 1.32 → 2.78 → 4.22
−1.67 → −1.12 → 0.34 → 1.77
−2.60 → −2.11 → −0.68
−3.57 −4.56 →
−3.12
probability of up movement: p = 1/2
−5.57
in position (i, j) = (2, 1), we can only go to 4.13 (an up movement) or to 1.68 (a down
movement). Although there is also the temptation to go to 6.58, this is not allowed in
the binomial tree. This latter interest rate (6.58%) can only be reached from 5.00% in the
earlier period. The tree in Table 10.10 shows this fact, as it also reports the arrows linking
the rates on the tree. However, these arrows are cumbersome, and so we will not report
them in future trees.
The simplicity of the model described in Equations 10.8 and 10.9 comes at a cost,
though. The model may generate negative interest rates, as is shown in the tree in Table
10.10. This is economically unreasonable: A negative interest rate means that investors are
willing to give $100 to the government today, in return for $90, for instance, in one year.
While negative real rates are a possibility, as inflation may be higher than nominal rates,
as occurred at the end of the 1970s, the interest rates in Table 10.10 are nominal. This is a
clear drawback of the model, and we will investigate models that overcome this problem.
However, we can anticipate that many interest rate models used daily by practitioners may
generate negative interest rates. We discuss this issue further in Chapter 11.
10.4.2
Risk Neutral Pricing
Obtaining the prices of derivative securities on a longer tree is no more difficult than
obtaining them on a short tree, such as the ones discussed in Chapter 9 and earlier in this
chapter. In particular, if we know the risk neutral probability at time i for an up movement,
then the formula of the price at time i in node j is
Vi,j
= e−r i , j ×Δ × E ∗ [Vi+1 ]
−r i , j ×Δ
= e
×
[p∗i
× Vi+1,j + (1 −
(10.12)
p∗i )
× Vi+1,j +1 ]
(10.13)
Therefore, if we know the value of the security at maturity (e.g., a zero coupon bond
must deliver $100) then we can move backward on the tree by using Equation 10.13. The
procedure is particularly easy to implement on a spreadsheet as the formula in Equation
370
MULTI-STEP BINOMIAL TREES
Table 10.11
Zero Coupon Bond Prices on January 8, 2002
Maturity (years)
Price
Yield
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
99.1338
97.8925
96.1462
94.1011
91.7136
89.2258
86.8142
84.5016
82.1848
79.7718
77.4339
1.74
2.13
2.62
3.04
3.46
3.80
4.04
4.21
4.36
4.52
4.65
Source: The Wall Street Journal.
10.13 can be entered into a given cell, and then copied and pasted in the remaining part of
the tree. Automatically, the spreadsheet delivers the outcome.
Where do we get the risk neutral probabilities p∗i ? One methodology is the one discussed
in the previous pages, namely, we use current zero coupon bond prices to compute p∗i ’s.
In particular, we can proceed recursively, meaning that we start from the bond expiring at
time i = 2 to compute p∗0 (the bond expiring at time i = 1 does not depend on risk neutral
probabilities, as its value is simply P0 (1) = e−r 0 ×Δ × 100), the bond expiring at time
i = 3 to compute p∗1 , and so on.
Table 10.11 provides the zero coupon prices on January 8, 2002, up to maturity T = 5
years.
We already know from previous sections that p∗0 = 0.6448 and p∗1 = 0.7869. We now
illustrate the procedure to find p∗2 . The other probabilities can be obtained in a similar
fashion. These are the steps:
1. Choose a value for p∗2 , for instance, p∗2 = 0.5, and build a four-step tree (the computer
does this for us) for the zero coupon bond with maturity i = 4. This is accomplished
by using the formula in Equation 10.13.
• The bond price we obtain will be in general different from the one in the data,
which is P0 (4) = 94.1011 in Table 10.11, because we are using a value of p∗i
chosen ad hoc. This fact is illustrated in Panel A of Table 10.12.
• Note that this tree, however, uses the correct risk neutral probabilities p∗0 =
0.6448 and p∗1 = 0.7869, as they were computed in the first two steps.
2. Once we have the tree, we can then search for p∗2 that matches the bond price in
the data. For instance, in Microsoft Excel, we can use the solver function. In this
function, we require the spreadsheet to find p∗2 so that the root of the tree (in the box)
equals the value of the bond in the data. The result of this procedure is illustrated in
Panel B of Table 10.12. The resulting risk neutral probability p∗2 = 0.6490.
MULTI-STEP TREES
Table 10.12
Two Trees for the 2-Year Bond
Panel A: Tree with Exogenously Specified p∗2 = 0.5
Price to Match
94.1011
time i
RN prob. p∗i
0
0.6448
1
0.7869
2
0.5
3
4
94.2732
93.8524
97.3565
94.9560
97.3063
99.7173
96.7635
97.9562
99.1635
100.3807
100
100
100
100
100
Panel B: Tree with Optimally Chosen p∗2
Price to Match
94.1011
time i
RN prob. p∗i
0
0.6448
1
0.7869
2
0.6490
3
4
94.1011
93.6811
97.1789
94.7827
97.1287
99.5360
96.7635
97.9562
99.1635
100.3807
100
100
100
100
100
371
372
MULTI-STEP BINOMIAL TREES
Table 10.13
The 5-Year Bond Tree
time =⇒
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
period i =⇒
0
1
2
3
4
5
6
7
8
9
10
RN prob. p∗i (%) 64.48 78.69 64.90 72.30 54.18 40.02 38.93 49.93 60.96 45.15
j
0
79.77 77.28 76.91 76.46 77.38 78.33 79.79 82.46 86.74 92.58 100
86.27 84.82 83.30 83.28 83.27 83.79 85.55 88.88 93.72 100
1
93.54 90.75 89.62 88.52 87.99 88.75 91.09 94.88 100
2
98.85 96.44 94.10 92.40 92.06 93.34 96.04 100
3
103.79 100.03 97.04 95.50 95.65 97.22 100
4
106.34 101.90 99.07 98.02 98.42 100
5
107.01 102.77 100.45 99.63 100
6
106.62 102.94 100.86 100
7
105.49 102.10 100
8
103.36 100
9
100
10
Table 10.13 shows the result of this procedure as we go forward to use all of the bonds
available in Table 10.11. In particular, it shows the tree for a 5-year zero coupon bond. We
are next going to use this tree in a couple of examples.
Given the tree in Table 10.13 and the risk neutral probabilities, we can now obtain the
price of any other interest rate security. The next section provides an application.
10.5
PRICING AND RISK ASSESSMENT: THE SPOT RATE DURATION
Numerous structured products contain implicit positions in options which make the security
sensitive to changes in interest rates. Risk managers must be able to:
1. Correctly evaluate the value of such embedded option.
2. Correctly evaluate the risk of the investment.
Although the two tasks are clearly related, they differ in an important respect. Pricing
(task 1) is performed by using risk neutral probabilities, as shown earlier. Risk assessment
(task 2) is performed by using risk natural, i.e., true, probabilities. Example 10.3 below
illustrates the difference. Before discussing the example, however, it is useful to introduce
a measure of interest rate risk that is similar to the notion of duration described in Chapter
3, Section 3.2 (see Equation 3.1). Because on a binomial tree the interest risk stems from
the variation of the spot rate, we call this risk measure spot rate duration, defined next:
Definition 10.1 The spot rate duration measures the percentage sensitivity of an interest
rate security price to the interest rate r and it is given by
D=−
dV
1
×
V
dr
(10.14)
PRICING AND RISK ASSESSMENT: THE SPOT RATE DURATION
373
How do we compute d V /dr on a binomial tree? We can use the values of the security
and the interest rate on the tree to approximate the variation in the price of the security
due to a variation in the interest rate (we use here the “u” and “d” notation, which is more
intuitive):
V1,u − V1,d
dV
≈
(10.15)
dr
r1,u − r1,d
EXAMPLE 10.3
A fund manager is offered a structured 5-year zero coupon bond, with the specific
characteristics that the total amount of principal repaid at time T = 5 is related to the
level of interest rates. Specifically, the bond pays at least 94% of the total principal
when interest rates are below 8.55%. When the interest rate increases above 8.55%,
the total principal paid increases proportionally with interest rates. The specific
payoff at time T = 5 (i = 10) is given by
Payoff at i = 10 : V10 = max(11 × 100 × r10 , 94)
(10.16)
The payoff at maturity of this bond is in Figure 10.1.
1. What is the fair value of this security?
The methodology developed in the previous pages provides an answer. We can
use the interest rate tree and the risk neutral probabilities that we computed to
price also this structured derivative (and many others). The binomial tree that
prices this security is contained in Table 10.14. The fair value is $79.88, which
is very close to the $79.77, the price of a standard zero coupon bond on the same
date (see Table 10.13). That is, the payoff at maturity is structured so that the
lower principal amount received when the interest rate is low is compensated
by a higher payoff when the interest rate is high.
2. What is the risk implicit in this security?
(a) First, we use the definition of spot rate duration in Equation 10.14 to compute
the sensitivity of the structured bond to changes in interest rates. From the
approximation in Equation 10.15 and the data in Tables 10.10 and 10.14, the
sensitivity to interest rate in the case of the structured bond is
D=−
79.14 − 83.19
V1,u − V1,d
1
1
×
= 2.08
×
=−
V0
r1,u − r1,d
79.88 3.39% − 0.95%
(10.17)
The same quantity computed for the standard five-year zero coupon bond (in
Table 10.13) yields D5 = 4.62, a much higher number. From this perspective,
therefore, the structured bond is less risky than the straight 5-year zero coupon
bond. The intuition is simple: Bond prices decline when interest rate increase.
The structured bond provides protection against this scenario, as it increases
the final payout when the interest rate increases. A similar protection could
be bought by purchasing some type of interest rate derivative. However, such
strategy would cost money today. The structured bond “pays” for the protection
374
MULTI-STEP BINOMIAL TREES
Figure 10.1
Payoff of Structured Zero Coupon Bond
180
160
140
Payoff
120
100
80
60
40
20
0
0
2
4
6
8
Interest Rate (%)
10
12
14
16
against an increase in the interest rate by reducing the principal repaid if the
interest rate instead declines.
(b) Second, we can evaluate the long-term payoff distribution of the security.
However, while for pricing we use risk neutral probabilities, for risk analysis
we must use the true, or risk natural, probabilities. The original tree with
the original p probabilities is therefore important in order to perform other
types of calculations that have nothing to do with pricing. The result is very
different depending on which probability we use. For instance, Table 10.15
shows the bond payoffs at maturity, and both the true probabilities and the risk
neutral probabilities to obtain such payoffs. While the risk neutral probabilities
assign 69.87% chance of ending up with the sub-par payoff of 94, the true
probabilities assign the much larger value of 82.81%. Conversely, while the
risk neutral probabilities assign over 30% to having a payoff above par (i.e.,
larger than 100), the true probabilities assign only around 17%. These results
should not be too surprising, as we already noted that risk neutral probabilities
make the expectation of future interest rates higher. Because this structured
security pays in a high-interest rate environment, it is overly optimistic. But
the point here is that any risk analysis, including Value-at-Risk and expected
shortfall discussed in Chapter 3, must be performed under the true probabilities.
Risk neutral probabilities are adjusted to take into account the market price of
risk, and therefore are distorted.
PRICING AND RISK ASSESSMENT: THE SPOT RATE DURATION
Table 10.14
i
j
0
1
2
3
4
5
6
7
8
9
10
375
5-year Structured Bond Tree
0
1
2
3
4
5
6
7
8
9
10
79.88
79.14
83.19
80.01
82.27
88.45
82.14
81.85
86.06
93.00
85.64
82.90
85.26
90.76
97.56
92.27
85.55
84.96
88.65
94.03
99.96
104.06
92.00
86.47
87.34
91.22
95.79
100.59
120.36
103.39
91.40
87.80
89.77
93.13
96.61
100.22
138.44
117.94
99.61
90.34
89.91
92.14
94.42
96.76
99.16
157.56
134.29
110.38
94.69
91.39
92.52
93.65
94.81
95.97
97.15
184.91
158.07
131.12
104.17
94.00
94.00
94.00
94.00
94.00
94.00
94.00
Table 10.15
Payoff at T = 5 of Structured Derivative
Payoff at T
True Probability
Risk Neutral Probability
184.91
158.07
131.12
104.17
94.00
0.10%
0.98%
4.39%
11.72%
82.81%
0.28%
2.35%
8.76%
18.74%
69.87%
376
10.6
MULTI-STEP BINOMIAL TREES
SUMMARY
In this chapter we covered the following topics:
1. Two-step binomial tree: This is a one-step extension of the the one-step binomial
tree. After a first upward movement in the interest rate, the interest rate can then
move again either upward or downward. Similarly, after a downward movement in
the interest rate, the interest rate can then move again either upward or downward.
There are four possible scenarios at time i = 2: (up, up), (up, down), (down, up),
(down, down). Most of the time, we choose nodes so that the interest rate after an
up/down movement equals the interest rate after a down/up movement. This is called
a recombining tree.
2. Risk neutral pricing: Given a risk neutral probability p∗ , the value of any security
equals the present value of the payoff discounted at the risk free rate. On a binomial
tree, this implies moving backward from the end of the tree, and at any node,
computing the discounted expected value.
3. Dynamic replication strategy: A portfolio of long-term and short-term bonds can
replicate the value of a derivative security along its binomial tree. The trading
strategy calls for the rebalancing of the portfolio at any node of the tree. The value
of the portfolio at time 0 equals the value of the derivative security.
4. Self financing strategy: The dynamic replicating strategy pays for itself, in the sense
that the replication strategy does not require any additional capital along the way.
5. Multi-step trees. Long-term binomial tree: We covered one specific way to construct
such a tree. Start from the forecast of future interest rates, and then consider
symmetric variations around the expected value, to obtain a simple recombining tree.
6. Risk neutral probabality computation: Use the current term structure of interest rates
to compute the risk neutral probability. By moving recursively from the shortest
maturity bonds to the longer maturity bonds it is possible to compute the probability
for every future period. This probabability can then be used to compute the value of
any other interest rate derivative.
7. Spot rate duration: The spot rate duration is the sensitivity of the interest rate security
to changes in the spot rate. The key is to approximate the value of the first derivative
(dV /dr) by the ratio of the difference in values of the security in the subsequent
nodes divided by the difference in spot rates in those nodes. It provides a measure
of risk of the security.
10.7
EXERCISES
1. Using the past history of short-term interest rates, you estimated by regression the
model
rt+dt = α + βrt + ut+dt
Suppose that the parameter estimates generated the tree for interest rates in Table
10.16, where there is equal probability to move up or down the tree. Assume also
EXERCISES
Table 10.16
i=0
377
An Interest Rate Tree
i=1
i=2
r2 , u u = 0.1
r1 , u = 0.07
r2 , u d
= 0.05
r2 , d u
r0 = 0.04
r1 , d = 0.03
r2 , d d = 0.02
for simplicity that each interval of time represents 1 year, that is, Δ = 1. Finally,
assume that the current zero coupon bond expiring at time i = 2 has a price equal to
Z0 (2) = 0.9
(a) How does the 2-year bond Z0 (2) evolve? Compute Z1,u (2) and Z1,d (2), and
draw the tree for the bond that expires at time i = 2 [recall the notation: Zi,j (k)
is the bond at time i in node j with maturity date k.]
(b) Use the calculation in Part (a) to compute the market price of risk λ embedded
in the current 2-year zero coupon bond Z0 (2) .
(c) Consider an option with maturity T = 1 to buy (at T = 1) one unit of a 1-year
zero coupon bond at the price of K = 95.
i. What is the market price of risk of this option? Why?
ii. Use your calculation in part i to compute the value of the option.
iii. Confirm your calculation by using the risk neutral approach.
(d) Assume that the risk neutral probabilities computed above are constant over
time. Compute the price of a 2-year European call option on a 1-year zero
coupon bond, with strike price K = .96.
(e) Compute the replicating portfolio that replicates the option payoff in Part (d).
Check that the portfolio indeed replicates the payoff. Discuss the intuition.
2. Consider the interest rate tree in Table 10.17. Assume that each interval of time
represents 1 year. All entries are continuously compounded interest rates. You
received mixed up information about the risk neutral and the risk natural (true)
probability of moving up the tree. You know it can only be one of the two cases
in Table 10.18. That is, in Case 1 the risk neutral probability is 70% and the true
probability is 30%. In Case 2, the risk neutral probability is 30% and the true
378
MULTI-STEP BINOMIAL TREES
Table 10.17
i=0
An Interest Rate Tree
i=1
i=2
r2 , u u = 0.09
r1 , u = 0.06
r2 , u d
= 0.04
r2 , d u
r0 = 0.04
r1 , d = 0.03
r2 , d d = 0.02
Table 10.18 Risk Neutral and Risk Natural Probabilities
Risk Neutral q
Risk Natural (True) p
Case 1
0.7
0.3
Case 2
0.3
0.7
probability is 70%. Only Case 1 or Case 2 is correct, but you do not know which
one. However, you know that a 2-year zero coupon bond costs Z(0, 2) = 91.31.
(a) Use the information provided to find the risk neutral probability of moving up
the tree, and compute the tree corresponding to a 3-year zero coupon bond.
(b) An investor buys the 2-year zero coupon bond at time i = 0. What is his/her
1-year expected return on the investment, as of i = 0? What if the trader buys
at i = 0 the 3-year zero coupon bond? What is his/her 1-year expected return
then?
(c) A range bond is structured security, which can be described as follows: It is
like a standard coupon bond, but it pays the coupon at some given time t if
the reference interest rate at time t − 1 is within a given interval (the range).
Otherwise, it pays no coupon at that time (it may pay it in the future, if the
condition is met). In any case, it will pay the principal at maturity T . Consider
a 3-year range bond, with a coupon equal to $10 / year. The range bond pays
the coupon at time i if the (continuously compounded) interest rate at time i − 1
is within the interval [r, r] = [0.025, 0.05].
i. Compute the value of the range bond at time i = 0.
ii. You are conducting a long-term risk analysis of this bond. Draw a histogram of the value of the bond at time i = 2 and compute the 9%
EXERCISES
379
Value-at-Risk (VaR) at time i = 2. For simplicity, in this computation,
simply compare the value of the bond at time i = 2 with the value of the
bond at time i = 0 (i.e. forget about the coupons).
iii. You can compute the 9% VaR by using either p or q above. Explain which
one you should use and discuss what difference you would get if you used
the wrong probability.
iv. What is the advantage of a range bond over a regular bond? Why do you
think these bonds became very popular around 1993? Explain.
3. Building a multi-step tree. Using quarterly past data on 3-month LIBOR (see Britsh
Bankers’ Association web site www.bba.org.uk), run the regression
rt+ 1 = α + βrt + t+1
where t ∼ N (0, σ 2 ).
(a) Use the fitted values α and β to compute the projected future interest rate
mt+i = E[rtoday +i ] (see Chapter 7).
(b) Use the estimated σ to define a binomial tree, as in Section 10.4.
(c) Using the swap rates for the same date, available at the Federal Reserve
Web Site (www.federalreserve.gov/Releases/h15/data.htm), compute the zero
coupon yield curve. Assume for simplicity that both fixed and floating rate
payments occur at quarterly frequency.
(d) Compute the risk neutral probabilities. Can you make sure the probabilities are
between 0 and 1? If not, try to increase the volatility σ used to fit the tree.
(e) Compute the expected risk neutral interest rate, and compare it with the predicted interest rate in Part (a). Discuss the difference.
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CHAPTER 11
RISK NEUTRAL TREES AND DERIVATIVE
PRICING
The methodology illustrated in Chapter 10 to build a binomial tree has some drawbacks.
For instance, there is no guarantee that the probability p∗i s that are obtained from matching
the term structure of interest rates are always between zero and one, as they should be. To
make sure that p∗i s are between these natural boundaries, we sometimes need to decrease
the step size Δ appropriately, a relatively cumbersome procedure.
To overcome this problem, the industry practice has moved to a different strategy,
namely, the construction of risk neutral trees without any reference to the true interest rate
tree. In this section, we review two popular risk neutral tree constructions, in which the
risk neutral probabilities are set equal to p∗ = 1/2, and the nodes of the tree are chosen
in a way consistent with the prices of interest rate securities. In addition, in this chapter
we extend the binomial tree methodology to price a wide variety of interest rate securities,
from coupon bonds to standard derivatives, such as caps, floors, and swaptions.
11.1
RISK NEUTRAL TREES
In this section we describe two popular interest rate models that are widely used to price
and hedge interest rate derivative securities.
11.1.1
The Ho-Lee Model
The Ho-Lee model is one of the simplest models that exactly fits the term structure of
interest rates. It is related to the model for interest rates we studied in Section 10.4 in
381
382
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Chapter 10. The model is specified as follows. First, let’s fix a time step, such as Δ = 0.5.
Let ri,j be the continuously compounded interest rate in node j between steps i and i + 1.
Then, for every (i, j) the Ho-Lee model postulates
ri+1,j
=
ri,j + θi × Δ + σ ×
ri+ 1,j + 1
=
ri,j + θi × Δ − σ ×
√
√
Δ
with RN probability p∗ = 1/2 (11.1)
Δ
with RN probability p∗ = 1/2 (11.2)
where recall from Chapter 10 that an upward movement in the binomial tree is characterized
by the node j remaining constant, and a downward movement by an increase in the node
j. As we showed in Chapter 10, this binomial tree is recombining.
What are the “θi ,” for i = 0, 1, ..., appearing in expressions (11.1) and (11.2)? These
are free parameters that are chosen to fit exactly the current term structure of interest rates.
Different levels of θi imply different nodes on the tree, while the risk neutral probability p∗
remain the same. As we did for risk neutral probabilities in Section 10.4 in Chapter 10, we
choose θ0 to exactly price the bond with maturity i = 2 (the bond with maturity i = 1 only
depends on r0 , and so it is independent of the location of r1,0 and r1,1 ); then we choose θ1
to exactly price the bond with maturity i = 3, and so on. An example of the methodology
will clarify the procedure. Recall that on multi-step trees we denote
Pi,j (k) = Bond price at time i in node j with maturity at (step) k
EXAMPLE 11.1
Consider the term structure of interest rates on January 8, 2002. The term structure
of interest rates and the zero coupon bonds are given in Table 10.11 in Chapter 10. In
the data, the zero coupon bond expiring on date k = 1 is P0 (1) = 99.1338, implying
r0 = 1.74%, which is the root of the tree.
θ0 : In the data, the zero coupon bond expiring on date k = 2 is P0 (2) = 97.8925. We
now choose θ0 so that the binomial tree exactly gives P0 (2) = 97.8925 as price.
From the model in Equations 11.1 and 11.2, we have
r1,0
=
1.75% + θ0 × Δ + σ ×
r1,1
=
1.75% + θ0 × Δ − σ ×
√
√
Δ
with RN probability p∗ = 1/2
Δ
with RN probability p∗ = 1/2
We first set σ as the volatility of interest rates, given by σ = 0.0173 according to the
data (see Chapter 10). Second, we can now choose θ0 so that the following equation
is satisfied
1
1
−r 0 ×Δ
−r 1 , 0 ×Δ
−r 1 , 1 ×Δ
×
e
×
e
=
e
×
+
× 100
97.8925
2
2
Price of zero in the data = Risk neutral price from binomial tree
Given r0 = 1.73% and σ = 0.0173, r1,0 and r1,1 both depend only on the level of
θ0 . Thus, we have one equation in one unknown. Using a search algorithm (e.g.
RISK NEUTRAL TREES
Table 11.1
Two Trees for a Zero Coupon Bond Expiring on k = 3
Price to Match
96.1462
θ1 = 0
Interest Rate Tree
1.74%
3.75%
1.30%
383
Optimal θ 1 = 0.021824
Interest Rate Tree
1.74%
3.75%
1.30%
4.97%
2.52%
0.08%
Zero Coupon Bond Price
96.6722
96.3241
97.5455
98.7098
98.7461
99.9614
100
100
100
100
6.06%
3.61%
1.17%
Zero Coupon Bond Price
96.1462
95.8000 97.0147
98.1727 98.2088
99.4175
100
100
100
100
solver in Microsoft Excel), we find θ0 = 1.5674%. Given this value for θ0 , the two
interest rates are r1,0 = 3.75% and r1,1 = 1.30%.
θ1 : In the data, the zero coupon expiring on date k = 3 has price P0 (3) = 96.1462.
Keeping θ0 as determined in the previous step, we now look for θ1 such that the tree
exactly yields a price P0 (3) = 96.1462. Rather than using an equation to find θ1 , we
use a different methodology, namely, the binomial tree itself. More specifically, let
us set up a three-step binomial tree for a given θ1 , e.g. θ1 = 0. This tree will provide
a bond value different from the one that we need. However, we can then vary θ1 until
we reach the correct value for the bond. Table 11.1 shows the result: On the left-hand
side there is an interest rate tree and bond price for the case in which θ1 = 0. On the
right-hand side of the table, instead, there is the interest rate tree and the bond price
for the θ1 that exactly matches the bond price in the data for maturity k = 3. As can
be seen comparing the two trees, the one on the right-hand side has nodes r2,0 ,r2,1
and r2,2 that are higher than the corresponding nodes on the left-hand side. That is,
θ1 had to be chosen greater than 0 to match the term structure of interest rates.
Moving on in this fashion, we obtain the risk neutral tree in Table 11.2 This risk
neutral tree exactly matches the term structure of interest rates.
11.1.2
The Simple Black, Derman, and Toy (BDT) Model
The main drawback of the binomial trees introduced earlier, both in Chapter 10 and above
in Section 11.1.1 is that it allows for negative interest rates. In this section we go over a
model that solves this problem. The model is specified as follows. For every time/node
(i, j) define the variable
zi,j = ln (ri,j )
384
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Table 11.2
Time T
Period i
θ i (×100)
j
0
1
2
3
4
5
6
7
8
9
10
The Risk Neutral Ho-Lee Interest Rate Tree
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0
1
2
3
4
5
6
7
8
9
1.5675 2.1824 1.4374 1.7324 0.7873 0.0423 -0.0628 0.4322 0.9271 0.1202
1.74
3.75
1.30
6.06
3.61
1.17
8.00
5.56
3.11
0.66
10.09
7.65
5.20
2.75
0.31
11.71
9.26
6.82
4.37
1.92
-0.52
12.95
10.51
8.06
5.61
3.17
0.72
-1.73
14.15
11.70
9.25
6.81
4.36
1.91
-0.53
-2.98
15.59
13.14
10.69
8.25
5.80
3.35
0.91
-1.54
-3.99
17.27
14.83
12.38
9.93
7.49
5.04
2.59
0.15
-2.30
-4.75
5
10
18.56
16.11
13.66
11.22
8.77
6.32
3.88
1.43
-1.02
-3.46
-5.91
The model specifies then that zi,j follows the process
√
zi+1,j = zi,j + θi × Δ + σ × Δ with RN probability p∗ = 1/2 (11.3)
√
zi+ 1,j + 1 = zi,j + θi × Δ − σ × Δ with RN probability p∗ = 1/2 (11.4)
This is the same model as Ho-Lee in Equations 11.1 and 11.2, but in logarithms. That is,
the model specifies a dynamic for the log of interest rates zi,j , which can be negative, but
the interest rate is then given instead by
ri,j = ez i , j ,
(11.5)
which is always positive.
As for the Ho-Lee model, the constants θi , for i = 1, 2, ... are determined to fit the term
structure of interest rates. This model is a special case of a more general model, developed
in Section 11.3 and called the Black, Derman and Toy model, in which also σ depends
on the time step i. To avoid confusion, we refer to the simpler version as “simple” BDT
model.
The next example illustrate the simple BDT model for the same set of bonds as in
Example 11.1.
EXAMPLE 11.2
The strategy to fit the term structure of interest rates is the same as for that of the
Ho-Lee model. Namely, we first look for θ0 that yields exactly the price of a bond
maturing on k = 2. Then, we move to find θ1 that fits exactly the price of the bond
maturing on k = 3. And so on. Without repeating the details of the procedure, as
they are the same as in Example 11.1, Table 11.3 shows the risk neutral tree. The
important detail to notice is the level of σ that we need to choose for the model.
Note that differently from the Ho-Lee model, now σ is the volatility of log-interest
rates zi = log(ri ). As such, it must be estimated from a log interest rate series.
Taking log differences in monthly interest rates from 1961/12 to 2001/12, we obtain
an (annualized) level of volatility equal to σ = 21.42%.
RISK NEUTRAL TREES
Table 11.3
time T
period i
θ i (×100)
j
0
1
2
3
4
5
6
7
8
9
10
11.1.3
385
The Risk Neutral Simple Black, Derman, and Toy Interest Rate Tree
0
0
71.82
0.5
1
69.16
1
2
33.48
1.5
3
33.79
2
4
11.82
2.5
5
-2.30
3
6
-4.38
3.5
7
4.55
4
8
12.81
4.5
9
-1.26
5
10
1.74
2.90
2.14
4.77
3.52
2.60
6.56
4.84
3.58
2.64
9.03
6.67
4.93
3.64
2.69
11.15
8.24
6.08
4.49
3.32
2.45
12.83
9.47
7.00
5.17
3.82
2.82
2.08
14.60
10.78
7.97
5.88
4.35
3.21
2.37
1.75
17.38
12.84
9.48
7.00
5.17
3.82
2.82
2.09
1.54
21.56
15.92
11.76
8.69
6.42
4.74
3.50
2.59
1.91
1.41
24.93
18.41
13.60
10.05
7.42
5.48
4.05
2.99
2.21
1.63
1.21
Comparison of the Two Models
By construction (i.e. choice of θi ’s), the two models discussed in the previous sections are
equally able to fit the current term structure of interest rates on January 8, 2002. However,
the two models generate important differences in the implied risk neutral probability distribution of interest rates in the future. To illustrate the differences, Figure 11.1 shows the
(smoothed) risk neutral distribution of interest rates at T = 5 (i.e. i = 10) under the two
models. The difference is apparent:
1. The Ho-Lee model gives non-zero probability to negative interest rates, and small
probability to high interest rates.
2. The Simple Black, Derman, and Toy model gives essentially zero probability to
interest rates below 1%, but it assigns a much higher probability to high interest
rates.
The type of probability distribution is in fact different: The Ho-Lee model generates a
symmetric, bell-shaped distribution of interest rates in the future, that looks like a normal
distribution. In contrast, the Simple Black, Derman and Toy model generates an asymmetric, positively skewed distribution of interest rates, that looks like a log-normal distribution.
In fact, as the time step Δ approaches zero, the distribution becomes normal in the case of
the Ho-Lee model and log-normal in the case of the Simple Black, Derman and Toy model
(see Chapter 14).
By construction, these differences are not important for bond prices, as both models
exactly match the term structure of interest rates. However, they will generate important
differences for other securities that have asymmetric payoff structures, such as options. As
a simple example, the structured zero coupon bond discussed in Section 10.5 in Chapter 10
has quite a different price depending on whether we use the Ho-Lee model or the Simple
Black, Derman and Toy model. Recall that the structured bond discussed in Section 10.5
has a payoff
Payoff of structured bond at T = max (11 × 100 × rT , 94)
386
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Figure 11.1
The Risk Neutral Distribution of Interest Rates at T = 5
0.25
Ho−Lee
Simple Black, Derman, and Toy
Probability Distribution
0.2
0.15
0.1
0.05
0
−10
−5
0
5
10
Interest Rate rT (%)
15
20
25
Using the same methodology used in Table 10.14, we find
Price of structured bond: Ho-Lee
Price of structured bond: Simple Black, Derman, and Toy
=
$80.0645
=
$78.9135
The lower price in the Simple Black, Derman and Toy model highlights the differences
of the model: Although the positive skewness of the risk neutral distribution in the BDT
model implies a higher risk neutral expected payoff for the Simple BDT model, the higher
interest rates implied by the model also imply a higher discount applied to the payoff. The
higher discount effect more than compensates for the higher expected return.
11.1.4
Risk Neutral Trees and Future Interest Rates
There is often a temptation to interpret too much from the implied risk neutral interest rate
trees. For instance, the Simple Black, Derman, and Toy model fitted in Table 11.3 implies
that from the current interest rate r0 = 1.74%, future interest rates can only go up. In fact,
both interest rates r1,u and r1,d are higher than the current interest rate. Although of course
this is a rather peculiar property of an interest rate process, we have to remember what a
risk neutral interest rate tree is. This is a tree whose only purpose is to compute the price
of interest rate securities through no arbitrage. The fact that the interest rate can only go
up in the model fitted in Table 11.3 has little to do with the real world expectation of future
interest rates, as we have to remember that a risk neutral tree embeds the risk aversion of
investors, as discussed in Chapter 10. In this model, risk aversion is embedded in the level
of θi . Lower risk aversion, for instance, would imply a lower level of θi and thus a better
behaved tree.
USING RISK NEUTRAL TREES
387
Having said so, however, the fitted Simple Black, Derman, and Toy model in Table
11.3 does expose a shortcoming of the model, and this is the fact that it does not allow
enough (risk neutral) probability mass to low interest rates. In contrast, the Ho-Lee model
in Table 11.2 allows perhaps too much (risk neutral) probability to low interest rates, and
in fact even to negative interest rates. Derivative security prices are very sensitive to this
distributional differences. The Simple Black, Derman and Toy model, as well as the full
BDT model discussed below, does not perform too well in low interest rate environments,
such as those in 2003 and, in fact, 2008. The reason is that to fit the term structure of
interest rates, it turns out that such models must give essentially zero probability to further
declines in interest rates, as in Table 11.3, a property that leads to a serious difficulty in
matching the prices of options, as discussed below.
11.2
USING RISK NEUTRAL TREES
In this section we illustrate the use of risk neutral trees to price other interest rate securities,
such as caps, floors, swaps and swaptions. First, we must learn how to include intermediate
cash flows.
11.2.1
Intermediate Cash Flows
Notice that given a tree, we can insert any type of known cash flow. Specifically, at any
time-node (i, j), we just must add the cash flow to the value to discount at time i + 1.
1
1
−r i , j ×Δ
Pi+ 1,j + Pi+1,j +1 + CF (i + 1)
×
(11.6)
Pi,j = e
2
2
where CF (i + 1) is the cash flow that will be paid at time i + 1.
EXAMPLE 11.3
Consider the price of a 1.5-year, 3% coupon bond on January 8, 2002. Let us use the
Simple Black, Derman, and Toy interest rate model fitted in Section 11.1.2. We can
calculate the price of the coupon bond using the tree in Table 11.4. In each step, we
add the cash flow CF (i + 1) = 1.5 (= 3% × 100/2) to the price in the following
period, and take the present value according to Equation 11.6. So, for example, the
value of the bond if the interest rate goes up twice (to r2,u u = 4.77%) is equal to the
present value of the bond value in the next period, equal to $100, plus the coupon to
be received next period, equal to $1.5. The present value is then P2,u u = $99.1094.
The prices on the tree are ex-coupon prices, that is, the price of the bond right after
the coupon has been paid.
11.2.2
Caps and Floors
A plain vanilla cap with maturity T , strike rate rK , and notional N is a security that pays
a stream of cash-flows at given dates T1 , T2 , ..., Tm = T , according to the formula
CF (Ti ) = Δ × N × max (rn (Ti − Δ) − rK , 0)
(11.7)
where n is the number of payments per year (e.g. n = 2), Δ = 1/n = Ti − Ti−1 is the
amount of time between payments and rn (T ) is a reference floating rate interest rate with
388
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Table 11.4
i= 0
t= 0
A Coupon Bond Tree with Maturity k = 3
i= 1
t = 0.5
i= 2
t= 1
i= 3
t = 1.5
P 3 , u u u (3) = 100
P 2 , u u (3)
= e −4 . 7 7 % / 2 (100 + 1.5)
= 99.1094
P 1 , u (3)
= e −2 . 9 0 % / 2
×[ 12 (99.1094 + 99.7287)
+ 1.5] = 99.4667
P 0 (3)
= e −1 . 7 4 % / 2
×[ 12 (99.4667 + 100.3780)
+ 1.5] = 100.5438
P 3 , u u d (3)
= 100
P 3 , u d u (3)
P 2 , u d (3)
= e −3 . 5 2 % / 2 (100 + 1.5)
= 99.7287
P 1 , d (3)
= e −2 . 1 4 % / 2
×[ 12 (99.7287 + 100.1886)
+ 1.5] = 100.3780
P 3 , u d d (3)
= 100
P 3 , d d u (3)
P 2 , d d (3)
= e −2 . 6 0 % / 2 (100 + 1.5)
= 100.1886
P 3 , d d d (3) = 100
USING RISK NEUTRAL TREES
389
compounding frequency n, such as the 6-month T-bill rate or LIBOR.1 Each individual
payment is called caplet. Note the temporal difference between the timing of the cash flow
payment, time Ti , and the timing of the interest rate that determines this cash flow, time
Ti − Δ. In other words, it is the interest rate at time Ti−1 that determines the cash flow at
time Ti .
Caps are very popular interest rate securities that offer insurance against the interest
rate increasing above to some level rK . They are often attached to floating rate bonds, for
instance, so as to limit the amount of interest the bond issuers have to pay. Indeed, as a
practical example, homeowners who finance the purchase of their homes through adjustable
rate mortgages, most likely they also buy a cap, as these floating rate mortgages typically
have a provision stating that the mortgage rate will never exceed a given maximum rate.
A plain vanilla floor is instead a security that pays the stream of cash flows:
CF (Ti ) = Δ × N × max (rK − rn (Ti − Δ) , 0)
(11.8)
In this case, the insurance is against a decrease in interest rates. While borrowers like
insurance against interest rate hikes, lenders like insurance against interest rate declines.
Some adjusttable rate mortgages, for instance, also define a minimum interest rate, which
can be thought of as a floor.
Pricing caps and floors is simple with trees, as cash flows are determined by the interest
rate itself. The only difficulty is the time difference between when the cash flow is
determined (Ti − Δ) and when the cash flow is paid (Ti ). In particular, it is convenient to
denote by CFi,j (k) the cash flow determined in node (i, j), that is, by the floating interest
rate ri,j , but paid at time k > i. According to the terms of the contract, if the tree time
step is equal to Δ = 1/n, then k = i + 1. In other words, while solving for the price of
the cap, we have to remember that the cash flow is not paid at time/node (i, j), when it is
determined by ri,j , but at time i + 1, independent of whether at time i + 1 the interest rate
increased or decreased.
More specifically, assuming that the time step of the tree is in fact Δ = 1/n, the cash
flow determined by the interest rate ri,j is
CFi,j (i + 1)
where
=
Δ × N × max (rn (i, j) − rK , 0)
rn (i, j) = n × er i , j ×Δ − 1
(11.9)
(11.10)
is the corresponding interest rate with compounding frequency equal to n. Given these
cash flows, we use the backward recursive formula to obtain the value of the cap along the
tree. That is, we have
Vi,j
= Value at time/node (i, j) of all cash flows at times k > i
1
1
Vi+ 1,j + Vi+1,j +1 + CFi,j (i + 1)
= e−r i , j ×Δ ×
2
2
(11.11)
(11.12)
The following example illustrates the methodology.
1 For
notational simplicity, in this chapter we suppress the maturity of the reference interest rate itself, i.e., we
denote rn (T ) = rn (T , T + Δ).
390
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Table 11.5 The Cash Flow Tree of a 1.5-Year Cap
i=0
t=0
i=1
t = 0.5
i=2
t=1
r2 , u u = 4.77%
r2 (2, uu) = 4.82%
CF2 , u u (3) = 1.162
r1 , u = 2.90%
r2 (1, u) = 2.92%
CF1 , u (2) = 0.210
r0 = 1.74%
r2 (0) = 1.75%
CF0 (1) = 0
i=3
t = 1.5
−→ paid here
−→ paid here
r2 , u d = 3.52%
r2 (2, ud) = 3.55%
CF2 , u d (3) = 0.526
−→ paid here
r2 , d d = 2.60%
r2 (2, dd) = 2.62%
CF2 , d d (3) = 0.059
−→ paid here
r1 , d = 2.14%
r2 (1, d) = 2.15%
CF1 , d (2) = 0
EXAMPLE 11.4
Consider the value on January 8, 2002, of a 1.5-years cap, with semi-annual payment
(n = 2, Δ = 0.5), and with strike rate rK = 3%. Let the notional N = 100. We
apply the same Simple Black, Derman, and Toy risk neutral tree computed above in
Table 11.3. We proceed in two steps:
1. Cash Flow Tree. The first step to obtain the price of the cap is to build a cash
flow tree, that is, a tree that defines the cash flow that is determined (not paid)
in a given node (i, j). Given Equation 11.9, we obtain the tree in Table 11.5.
The cash flow tree in this table also shows not only the time of the formation of
the cash flows, but also when they would be paid (i.e., one period later). For
instance, if the interest rate increases twice, it ends up in r2,u u = 4.77%. The
corresponding semi-annually
compounded
interest rate, from Equation 11.10, is
r2 (2, uu) = 2 × e4.77% /2 − 1 = 4.82%. Thus, the cash flow determined at
time/node (2, uu) is C2,u u (3) = 100/2 × max (4.82% − 2.5%, 0) = 1.162. Note,
however, this cash flow is not paid at time (2, uu) but at time i = 3, as the tree shows.
2. Cap Value Tree. Given the cash flow tree, we can compute the value of the cap by
using the backward formula in Equation 11.12. The resulting tree is in Table 11.6.
We obtain a value of the cap at time i = 0 given by V0 = $0.647.
Table 11.6
i=0
t=0
The 1.5-Year Cap Value Tree
i=1
t = 0.5
i=2
t=1
i=3
t = 1.5
V2 , u u =
= e4 . 7 7 % / 2 × 1.162 = 1.135
CF2 , u u (3) = 1.162
V2 , u d =
= e−3 . 5 2 % / 2 × 0.526 = 0.517
CF2 , u d (3) = 0.526
V2 , d d =
= e−2 . 6 0 % / 2 × 0.059 = 0.058
CF2 , d d (3) = 0.058
V1 , u = e−2 . 9 0 % / 2 ×
×[ 12 (1.135 + 0.517) + 0.210]
= 1.021
V0 = e−1 . 7 4 % / 2 ×
×[ 12 (1.021 + 0.285) + 0]
= 0.647
USING RISK NEUTRAL TREES
V1 , d = e−2 . 1 4 % / 2 ×
×[ 12 (0.517 + 0.058) + 0]
= 0.285
391
392
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Table 11.7
A 5-Year Cap
Panel A: Cash Flow Tree
Period i
Node j
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0.00
0.21
0.00
1.16
0.53
0.06
2.08
1.20
0.55
0.08
3.37
2.14
1.25
0.59
0.10
4.48
2.95
1.84
1.02
0.42
0.00
5.37
3.60
2.31
1.37
0.68
0.17
0.00
6.32
4.29
2.81
1.74
0.95
0.37
0.00
0.00
7.83
5.38
3.61
2.31
1.37
0.68
0.17
0.00
0.00
10.13
7.04
4.81
3.19
2.01
1.15
0.52
0.05
0.00
0.00
Panel B: Cap Value Tree
Period i
Node j
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
9.44
12.19
6.86
15.10
9.21
4.64
17.35
11.25
6.45
2.84
18.93
12.76
7.89
4.13
1.46
19.44
13.42
8.68
5.00
2.24
0.52
18.89
13.25
8.82
5.40
2.79
0.92
0.12
17.29
12.25
8.33
5.32
3.03
1.30
0.23
0.01
14.33
10.22
7.06
4.65
2.83
1.47
0.44
0.03
0.00
9.10
6.50
4.53
3.05
1.95
1.12
0.51
0.05
0.00
0.00
This approach can be extended without any difference to any maturity, and to higher
frequency. Table 11.7 provides a 5-year cap with semi-annual payments and cap rate equal
to rK = 2.5%. Panel A reports the cash flow tree (at time/node of cash flow formation)
while Panel B reports the value tree. Note that the table only shows periods until i = 9 as
the last cash flow, in Period i = 9, is in fact paid at time i = 10, i.e. in five years.
11.2.3
Swaps
Recall our discussion of interest rate swaps in Section 5.4 in Chapter 5. The valuation of
this type of interest rate security can be simply obtained from the discount factors Z(0, T ).
However, the last two decades have seen the development of a large number of interest rate
securities whose payoffs depend on the value of interest rate swaps. Thus, understanding
the dynamics of the value of the interest rate swap on an interest rate tree is instrumental to
obtaining the price of these other interest rate derivatives. We now apply the risk neutral
USING RISK NEUTRAL TREES
393
tree methodology to value interest rate swaps. On top of these interest rate swaps, then, we
will cover a popular interest rate derivative, called swaptions.
The cash flow from a plain vanilla swap is
CF (Ti ) = Δ × N × (rn (Ti − Δ) − c)
(11.13)
where, recall, n is the number of payments per year (e.g. n = 2), Δ = 1/n is the amount
of time between payments, c is the swap rate, and rn (T ) is a reference floating rate interest
rate, such as the LIBOR, with n compounding frequency. The methodology to value the
interest rate swap is identical to the one discussed earlier for caps. Briefly, we follow two
steps:
1. Compute a cash flow tree using
CFi,j (i + 1) = Δ × N × (rn (i, j) − c)
where, recall, rn (i, j) = n × er i , j ×Δ − 1
(11.14)
2. Compute the value of the swap on the tree as the present value of the risk neutral
expectation of future cash flows by moving backward on the tree:
1
1
Vi+ 1,j (k, c) + Vi+1,j +1 (k, c) + CFi,j (i + 1)
Vi,j (k, c) = e−r i , j ×Δ ×
2
2
where
Vi,j (k, c) = Value of the swap in (i, j) with maturity (k), and swap rate c (11.15)
The next example illustrates the methodology.
EXAMPLE 11.5
Consider a 5-year fixed-for-floating swap on January 8, 2002, defined on the 6-month
T-bill rate and with semi-annual payments.2 According to Equation 5.43 in Chapter
5, the swap rate is
1 1 − Z(0, 10)
= 4.49%
(11.16)
c=
2 10
i= 1 Z(0, i)
Recall that this swap rate is the one that makes the value of the interest rate swap
equal to zero at inception. Given the Simple Black, Derman and Toy tree in Table
11.3, we obtain the cash flow tree and swap value tree in Panels A and B of Table
11.8, respectively. The comforting fact is that the root of the swap value tree, in Panel
B, indeed gives V0 = 0, as it should be from the definition of c. This is not surprising,
as the tree used to value this swap was calibrated to zero coupon bonds. However, we
have confirmation that the tree methodology works, as it correctly values the interest
rate swap.
2 As
discussed in Chapter 5, the most popular interest rate swaps have floating rates linked to the LIBOR rate.
However, as over-the-counter contracts, interest rate swaps are defined on most floating rates, including T-bill
rates, commercial paper rate, and others.
394
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Table 11.8
A 5-Year Swap Tree
Panel A: Cash Flow Tree
Period i
Node j
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
-1.37
-0.78
-1.17
0.17
-0.47
-0.93
1.09
0.21
-0.44
-0.91
2.38
1.15
0.25
-0.41
-0.89
3.49
1.96
0.84
0.03
-0.57
-1.01
4.38
2.61
1.32
0.37
-0.32
-0.82
-1.20
5.33
3.30
1.82
0.74
-0.05
-0.63
-1.05
-1.36
6.83
4.39
2.61
1.32
0.38
-0.31
-0.82
-1.20
-1.47
9.14
6.04
3.81
2.20
1.02
0.15
-0.48
-0.94
-1.28
-1.54
Panel B: Swap Value Tree
Period i
Node j
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0.00
4.27
-1.53
8.18
2.04
-2.79
11.38
5.04
0.05
-3.83
13.86
7.48
2.44
-1.47
-4.47
15.22
9.03
4.14
0.36
-2.54
-4.74
15.50
9.71
5.18
1.67
-1.00
-3.02
-4.55
14.72
9.58
5.58
2.51
0.18
-1.58
-2.90
-3.89
12.58
8.42
5.21
2.77
0.92
-0.46
-1.50
-2.27
-2.85
8.20
5.58
3.60
2.10
0.99
0.15
-0.47
-0.93
-1.27
-1.52
USING RISK NEUTRAL TREES
11.2.4
395
Swaptions
Recall from Chapter 6, Section 6.2, that a swaption, or option on a swap, is an interest rate
contract between two counterparties in which one counterparty (the option buyer) has the
right, but not the obligation, to enter at a prespecified time T into a given interest rate swap
with maturity T sw ap > T and (strike) swap rate rK . The other counterparty (the option
seller) has the obligation to take the other side of the swap contract if the option buyer
exercises the option.
Two main types of plain vanilla swaptions are the following:
1. A receiver swaption is an option to enter into a swap and receive the fixed rate rK .
2. A payer swaption is an option to enter into a swap and pay the fixed rate rK .
As with any option, swaptions provide insurance. In the case of swaptions the insurance
to the option buyer is against movements in interest rates. For instance, consider a company
that has issued a floating rate bond. The potential future liabilities from issuing a floating
rate bond is high, because if the floating rate increases, then the company has to make large
interest rate payments. A payer swaption at a given strike rate rK allows the company to
insure against increases in the floating interest rate. If the floating rate is high at maturity,
the company can exercise its option to enter into a fixed-for-floating swap, in which it pays
fixed rate rK and receives the floating rate. After the company exercises the option, it
effectively changes its liability into a fixed rate bond, as the floating rate it receives from the
swap can be used to hedge the floating rate payments the company must make to service
its debt.
As a second example, swaptions are also very popular hedging instruments for investors
in callable securities, such as callable bonds or mortgage backed securities.3 For instance,
an investor in callable bonds is worried about prepayment risk, namely, the risk that if the
interest rate declines, the issuer of the bond will call back the bond, and the investor will
receive his capital back too early. The investor then not only does not receive any more
coupons, but because the current interest rate is low, his investment opportunities probably
deteriorated, and so he or she won’t be able to invest the capital at the same attractive rates
of the original callable bond (being called). To hedge against this possibility the issuer can
buy a receiver swaption at a rate rK close to the original coupon of the callable bond. If the
interest rate declines and the bond is called, then, the investor can exercise the option and
receive the fixed rate payments instead of floating rates. The investor can also readily meet
the floating rate payments of the swap, as he can invest the capital he received back from
the issuer of the original callable bond in any instrument that pays a floating rate, such as a
money market account. Effectively, the investor can still receive the fixed rate rK rate on
the capital, as if the callable bond hadn’t been called.
How do we value the swaption premium today? The interest rate tree methodology
discussed earlier provides the answer. It is convenient to provide the methodology through
the discussion of an example:
3 In
fact, American swaptions are even better as hedging instruments. We cover them in Chapter 12.
396
RISK NEUTRAL TREES AND DERIVATIVE PRICING
EXAMPLE 11.6
Consider a European payer swaption with two years to maturity (i = 4), to enter at
i = 4 into a 3-year swap and pay the fixed rate rK = 4.49%. The maturity date of
the swap is then five years from now, i.e. k = 10.
Let ci,j (k) be the swap rate at time/node (i, j) for a swap maturing at time k.4
Intuitively, then, the buyer of a payer swaption will exercise the option if and only if
the current swap rate at maturity i = 4 is higher than the strike swap rate rK . That
is, exercise if and only if c4,j (10) > rK . The reason is that if the current swap rate is
high, the holder of the option has the opportunity to exercise the option and pay the
lower amount rK rather than the current swap rate c4,j (10). Vice versa, if the current
swap rate is low, i.e. c4,j (10) < rK , then it is more convenient for the option holder
to give up the option and enter into the payer swap at the current (lower) swap rate.
All this sounds complicated, as it appears we must first compute the dynamics
of the swap rate ci,j (10) before we can solve for the value of the swaption value.
Instead, we must only remember how ci,j (10) is defined. In particular, recall that the
swap rate ci,j (10) is that rate that makes the swap value at time (i, j) equal to zero.
That is, using the notation in (11.15), ci,j (10) is such that
Vi,j (10, ci,j (10)) = 0.
(11.17)
As a consequence, if the current market swap rate at time i = 4 and node j is above
the strike rate, c4,j (10) > rK , then the value of the swap with swap rate rK must be
greater than zero (which from Equation 11.17 is the value of the swap at the current
market swap rate). That is,
c4,j (10) > rK
if and only if
V4,j (10, rK ) > 0
(11.18)
This fact enables us to easily obtain the value of the swaption. In fact, Equation 11.18
implies that the value of the swaption at the option maturity i = 4 is
Payoff of swaption at time i = 4 : max (V4,j (10, rK ), 0)
(11.19)
In our example, rK = 4.49%, which is the same as the swap rate used in the swap tree
in Table 11.8. Thus, V4,j (10, rK ), for j = 0, 1, . . . 4 are given by the fifth column in
the table. For instance, V4,0 (10, rK ) = 13.86, V4,1 (10, rK ) = 7.48 and so on. The
payoff from entering into the swap is then given by the swap value in node j = 0,
j = 1 and j = 2, and zero otherwise. Given the final payoff of the swaption, we then
obtain the value at time i = 0 by solving the tree backward, as in the tree in Table
11.9. The price of the swaption is $3.41.
To summarize the methodology, Example 11.6 illustrates the fact that three steps are
necessary to price a European swaption on a binomial tree:
1. Compute the tree of the underlying swap value whose swap rate is equal to the
swaption’s strike rate rK .
2. Compute the swaption payoff as in Equation 11.19.
3. Use the risk neutral binomial tree to compute the price of the swaption from its
payoff.
4 Note
that the swap rate moves over time on the tree, as the short-term interest rate does.
IMPLIED VOLATILITIES AND THE BLACK, DERMAN, AND TOY MODEL
397
Table 11.9 A 2-Year Payer Swaption
i
j
0
1
2
3
4
11.3
0
1
2
3
4
3.41
5.11
1.76
7.41
2.97
0.59
10.33
4.84
1.20
0.00
13.86
7.48
2.44
0.00
0.00
IMPLIED VOLATILITIES AND THE BLACK, DERMAN, AND TOY MODEL
In our earlier examples on the pricing of interest rate caps, floors, and swaptions, we used
the empirical volatility σ as input in the interest rate tree in Table 11.3. That is, σ was
computed as the standard deviation of past realized changes in the short term interest rates.
Given the value of σ, we computed the tree that fitted exactly the term structure of interest
rates. However, it is an empirical regularity that such a value of σ typically underprices
caps, floors and swaptions. That is, the binomial trees built using the empirical interest rate
volatility give a valuation of standard options that is typically too low.
It has become standard practice in the industry, then, to compute σ not out of the past
changes in the interest rates rt , but rather straight from the traded prices of caps, floors, and
swaptions. In other words, the value of σ is chosen in a way that prices caps and floors.
Once σ is chosen, the term structure of interest rates is also fitted. This volatility level is
called implied volatility.
Definition 11.1 The empirical volatility of interest rates is the level of interest rate
variation σ computed from a time series of past interest rate changes. For instance, in the
two models discussed in Section 11.1, the empirical volatility is
Ho-Lee Model
Simple Black, Derman, and Toy Model
: σ = st.dev. of (rt+Δ − rt )
(11.20)
: σ = st.dev. of (ln(rt+Δ ) − ln(rt )) (11.21)
We now define the option’s implied volatility. We provide the definition for caps, but an
equivalent definition holds for any option’s contract.
Definition 11.2 Consider a given derivative security, such as a cap, with maturity T and
strike rate rK , and let capD ata (T, rK ) be the current price level of the cap. The implied
volatility of this cap is the level of interest rate variation σ such that the chosen interest
rate model yields a price of the cap identical to the capD ata (T, rK ).
We now illustrate these concepts using an example. The important caveat, though, is
that market data on caps, floors, and swaptions are readily available in case the underlying
floating rate is the LIBOR interest rate, as opposed to the Treasury bill rate used in the
previous sections. Thus, to be consistent between the term structure of interest rates and
the value of these derivative securities, we must fit the interest rate tree to the term structure
398
RISK NEUTRAL TREES AND DERIVATIVE PRICING
of interest rates implied by the LIBOR rate. Plain vanilla swaps, for instance, are LIBORbased, and thus we use the discount curve Z(0, T ) that is implied by swap rates. (The
methodology to extract Z(0, T ) from swap rates was illustrated in Section 5.4 in Chapter
5. Section 6.5 in Chapter 6 also discusses the use of Eurodollar futures to construct the
LIBOR curve.)
Table 11.10 contains the swap rates c(0, T ), the implied discount factors Z(0, T ), and
the cap prices on November 1, 2004.5 The swaps and the caps quoted in this table are for
quarterly payments. For each cap, the corresponding strike rate rK is given by the swap
rate with the same maturity. A cap with such a strike rate is said to be at-the-money. The
last two columns in Table 11.10 contain the prices of the same caps computed from the
Simple Black, Derman and Toy and Ho-Lee interest rate tree model when each is fitted to
the discount factors Z(0, T ) in column four.
Consider, for instance, the 1-year cap, whose value in the data is V data (0, 1) = 0.1859
and the strike rate is rK = c(0, 1) = 2.555%. Using the Simple Black, Derman, and Toy
model with the empirical volatility of LIBOR rates, i.e. σ = 15.06%, the resulting interest
rate tree is shown in Panel A of Table 11.11. Panel B shows the tree for a 1-year zero coupon
bond. The price P m odel (0, T ) = 97.4834 indeed corresponds to the zero coupon bond price
in Table 11.10. Panel C of Table 11.11 contains the cash flow tree for the cap, and Panel D the
cap value tree. As can be seen, the value is V m odel (0, 1) = $0.1520, which is substantially
lower than the corresponding value in the data, namely, V data (0, 1) = $0.1859.
The final two columns in Table 11.10 show the results of similar exercises performed
for all maturities for both the Simple BDT and Ho-Lee model with constant volatilities
(σ = 15.06% for the Simple BDT, and σ = 0.8326177% for Ho-Lee). As can be see, both
the Simple BDT and Ho Lee model with constant empirical volatility miss the cap prices
for all of the maturities.
One interesting finding in Table 11.10 is that the Ho-Lee model appears to overprice
short term caps, and underprice long term caps, while the Simple BDT model in this case
always underprices. Note that we used exactly the same data to estimate both models,
namely, the realized 3-months LIBOR rates for the 1987 to 2004 period, and the swap rates
in the second column of Table 11.10. The different performance of the two models, Simple
BDT versus Ho-Lee, stems from the different implied (risk neutral) probability distribution
of interest rates, as discussed in Section 11.1.3.
11.3.1
Flat and Forward Implied Volatility
One possible problem with the model is that the volatility σ has been mis-measured. After
all, the volatility of interest rates is time varying, and thus we may be using the wrong
level of volatility. The biggest problem, however, is that a single value of σ that makes
the observed cap price consistent with the model does not exist. Table 11.12 reports the
cap prices across maturities for various volatility levels. More specifically, the second
column reports the cap prices in the data. The next four columns report the cap prices
implied by the Simple Black Derman and Toy model for four levels of volatility σ = 0.188,
5 As
discussed in Chapter 20, caps, floors, and swaptions are in fact quoted in implied volatility units by dealers.
The implied volatility is from the Black model of interest rates, which we discuss in Chapters 20 and 21. The
dollar prices contained in Table 11.10 are computed out of the original implied volatility quotes using such a
model.
IMPLIED VOLATILITIES AND THE BLACK, DERMAN, AND TOY MODEL
Table 11.10
399
Swap Rates and Cap Prices on November 1, 2004
Maturity
T
Swap Rates
c(0, T )(%)
Discount
Z(0, T )
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
2.1800
2.3177
2.4420
2.5550
2.6586
2.7546
2.8451
2.9320
3.0167
3.0991
3.1784
3.2540
3.3254
3.3930
3.4577
3.5200
3.5805
3.6393
3.6962
3.7510
99.4580
98.8510
98.1899
97.4834
96.7385
95.9598
95.1503
94.3109
93.4417
92.5456
91.6268
90.6899
89.7397
88.7778
87.8050
86.8212
85.8263
84.8218
83.8102
82.7938
Original Data Source: Bloomberg.
—————- Cap Prices —————
Data
Simple BDT Model
Ho Lee Model
–
0.0456
0.1059
0.1859
0.2887
0.4157
0.5662
0.7364
0.9201
1.1129
1.3126
1.5194
1.7352
1.9598
2.1916
2.4288
2.6691
2.9117
3.1562
3.4029
–
0.0400
0.0948
0.1520
0.2106
0.3038
0.3984
0.4982
0.6062
0.7229
0.8586
0.9961
1.1386
1.2838
1.4344
1.5889
1.7542
1.9208
2.0954
2.2706
–
0.0689
0.1512
0.2349
0.3366
0.4457
0.5670
0.7050
0.8485
1.0008
1.1579
1.3252
1.4911
1.6643
1.8415
2.0247
2.2129
2.4007
2.5946
2.7889
400
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Table 11.11
LIBOR-Based Interest Rate Tree and the 1-Year Cap Value
Time t
Period i
0
0
0.25
1
0.5
2
0.75
3
1
4
Panel A: Interest Rate Tree
j
0
1
2
3
2.17%
2.63%
2.26%
3.10%
2.67%
2.30%
3.59%
3.09%
2.66%
2.29%
Panel B: Zero Coupon Bond
j
0
1
2
3
4
97.4834
97.8669
98.1623
98.4023
98.6241
98.8153
99.1064
99.2308
99.3380
99.4303
Panel C: Cap Cash Flow Tree
j
0
1
2
3
0.0000
0.0216
0.0000
0.1400
0.0308
0.0000
0.2629
0.1364
0.0277
0.0000
Panel D: Cap Value Tree
j
0
1
2
3
0.1520
0.2434
0.0622
0.3353
0.1114
0.0137
0.2606
0.1353
0.0275
0.0000
100
100
100
100
100
IMPLIED VOLATILITIES AND THE BLACK, DERMAN, AND TOY MODEL
Table 11.12
Cap Implied Volatilities: The Simple BDT and Ho-Lee Models
SIMPLE BDT
Maturity
Data
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
0.0456
0.1059
0.1859
0.2887
0.4157
0.5662
0.7364
0.9201
1.1129
1.3126
1.5194
1.7352
1.9598
2.1916
2.4288
2.6691
2.9117
3.1562
3.4029
401
HO-LEE
Implied Volatilities σ for Caps with Maturity T (in Parenthesis)
0.188
0.2291 0.30277 0.28504
0.00458 0.00584 0.01006 0.010997
(T = 0.5) (T = 1) (T = 3) (T = 5)
(T = 0.5) (T = 1) (T = 3) (T = 5)
0.0456
0.1061
0.1674
0.2385
0.3374
0.4428
0.5583
0.6790
0.8154
0.9577
1.1156
1.2751
1.4442
1.6160
1.7959
1.9791
2.1717
2.3696
2.5712
0.0518
0.1184
0.1859
0.2715
0.3735
0.4958
0.6258
0.7648
0.9202
1.0819
1.2543
1.4373
1.6291
1.8252
2.0283
2.2418
2.4639
2.6889
2.9181
0.0628
0.1398
0.2262
0.3324
0.4542
0.5944
0.7492
0.9274
1.1118
1.3115
1.5194
1.7454
1.9749
2.2170
2.4643
2.7226
2.9931
3.2734
3.5562
0.0602
0.1347
0.2166
0.3178
0.4336
0.5710
0.7182
0.8884
1.0640
1.2559
1.4528
1.6709
1.8895
2.1200
2.3560
2.6078
2.8646
3.1329
3.4029
0.0456
0.1047
0.1644
0.2224
0.3105
0.4053
0.4988
0.5996
0.7077
0.8256
0.9508
1.0740
1.2010
1.3288
1.4606
1.5976
1.7361
1.8781
2.0193
0.0534
0.1204
0.1859
0.2601
0.3510
0.4544
0.5651
0.6811
0.8005
0.9297
1.0625
1.2044
1.3457
1.4905
1.6387
1.7885
1.9432
2.0986
2.2597
0.0796
0.1727
0.2723
0.3899
0.5128
0.6594
0.8076
0.9697
1.1441
1.3287
1.5194
1.7071
1.9021
2.1047
2.3170
2.5288
2.7433
2.9592
3.1818
0.0854
0.1843
0.2926
0.4187
0.5522
0.7094
0.8668
1.0353
1.2233
1.4213
1.6245
1.8265
2.0375
2.2501
2.4777
2.7025
2.9337
3.1670
3.4029
0.2291, 0.30277, and 0.28504. Each level of volatility σ was chosen in such a way that the
Simple BDT model exactly prices one given cap, whose maturity is reported underneath
the volatility level. For instance, the first volatility level σ 6−m o = 0.2291 exactly prices
the cap maturing in six months, which is emphasized by the presence of a box around the
cap price level. The second volatility level σ 1−y r exactly prices the cap with one year to
maturity. And so on. Note that for each choice of σ, the interest rate tree, and thus θi , has
to be refitted to the current zero coupon yield curve.
The results in this table clearly show that even if the Simple BDT model with constant
volatility σ was able to exactly price one given cap, it would fail to price caps with different
maturities. The same comment applies to the Ho-Lee model.
The exercise in Table 11.12 also illustrates the notion of an implied volatility. For
instance, σ 6−m o = 0.188 is the implied volatility of a cap with maturity of six months, and
strike price rK = 2.3177 (= swap rate from Table 11.10). Similarly, σ 5−y r = 0.28504
is the implied volatility of a 5-year cap, with strike rate rK = 3.7510. And so on. In the
industry, these types of implied volatility are referred to as flat volatility, because a single
number characterizes the price of a given cap. The term flat is temporal in meaning: The
same volatility applies for every period in the future for that particular cap. However, this
notion of implied volatility is merely a quoting convention to characterize a given cap with
402
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Figure 11.2 Implied (Flat) and Forward Volatility on November 1, 2004
34
Implied (Flat) Volatility
Forward Volatility
32
30
Volatility (%)
28
26
24
22
20
18
16
0
1
2
3
4
5
Cap Maturity
6
7
8
9
10
Original Data Source: Bloomberg.
one number. Taken literally, in fact, it leads to a contradiction, as we discuss in the next
subsection.
Definition 11.3 The implied flat volatility of an interest rate cap with maturity T and
strike rate rK is the level of σ(rK , T ) in the interest rate model that exactly prices the cap.
One interesting fact we can gather from the data in Table 11.12 is that the Simple BDT
implied (flat) volatility first increases and then decreases with time to maturity. For instance,
σ 6−m o = 0.1880, but it then increases for maturity of 1 year and 3 year (σ 1−y r = 0.2291
and σ 3−y r = 0.30277), and then declines again to σ 5−y r = 0.28504. The solid line in
Figure 11.2 shows the implied flat volatility curve on November 1, 2004 for cap maturities
up to ten years.
11.3.2
Forward Volatility and the Black, Derman, and Toy Model
The tree that uses the implied volatility of a cap with maturity T (e.g. one year) and strike
rate rK (e.g., 2.4420%) is consistent with the current term structure of interest rates (swap
rates) and this particular cap. However, the fact that different maturities / strike prices need
different implied volatilities generates a logical inconsistency for the risk neutral pricing
methodology. Recall from Chapter 9 that risk neutral pricing stems from the possibility of
replicating one given interest rate security with another interest rate security (e.g. the 1year zero coupon bond with the 2-year zero coupon). Different implied volatilities generate
different trees. This suggests that it is not possible to replicate, for instance, the 1-year cap
by using the 2-year cap. According to the theory, if it is not possible to replicate, there
IMPLIED VOLATILITIES AND THE BLACK, DERMAN, AND TOY MODEL
403
must be an arbitrage opportunity. All this is not satisfactory. Instead, we would like to
have an interest rate model that is able to fit exactly all of the zero coupon bonds and all of
the caps. The full Black, Derman and Toy model is such a model. The key step is to free
up the volatility σ, and let it take different values in the future, i.e. set σ i for i = 1, 2, ....
One difficulty with simply adding a time index i to σ in the Simple BDT model in
Equations 11.3 and 11.4 is that the tree is no longer recombining. In fact, the log rate
zi,j = log(ri,j ) after an up and down movement is different from the log rate after a down
and up movement. Indeed, it is simple to verify that
√
z2,u d = z0 + (θ0 + θ1 ) × Δ + (σ 1 − σ 2 ) × Δ
√
z2,du = z0 + (θ0 + θ1 ) × Δ − (σ 1 − σ 2 ) × Δ
which differ from each other, unless σ 1 = σ 2 .
The Black, Derman and Toy model retains the spirit of the Simple BDT model (which
as we have noted is a special case), but resolves the issue by using a different procedure to
construct the tree. To introduce this procedure, it is convenient to see it first in the context
of the simple model presented in Equations 11.3 and 11.4. In this model, note first that for
every i we can write
√
(11.22)
zi,j + 1 = zi,j − 2 × σ × Δ for j = 0, 1, ..., i − 1
We can see this by simply subtracting Equation 11.4 from Equation 11.3 and rearranging
√
terms. For instance, according to
√ the model, the first step is z1,u = z0 + θ0 × Δ + σ Δ
from the second, we
and z1,d = z0 + θ0 × Δ − σ × √Δ. Taking the difference of the first√
obtain z1,d − z1,u = −2 × σ × Δ and thus z1,d = z1,u − 2 × σ × Δ.
The implication of Equation 11.22 is that instead of searching for θi at any step i, we
can instead search for zi+ 1,0 , that is, the top element in the interest rate tree. In fact, for
given value zi+1,0 , all of the remaining log rates zi+1,j +1 can be computed using Equation
11.22. To put it differently, the top interest rate zi+1,0 takes the place of θi .
The interesting aspect of this alternative but equivalent construction of the Simple BDT
model in Equation 11.3 and 11.4 is that we can now extend it to have different step-by-step
volatility, σ i , but the model still generates a recombining tree. This is exactly the Black,
Derman, and Toy model. In every step i, then, we look for two values, the top interest rate
on the tree zi,0 , and the volatility level σ i , in a way that the model contemporaneously fits
the zero coupon bond and the cap with maturity (period) i + 1. Given that at each stage
we have two equations in two unknowns, we are in a good position to be able to obtain
a unique tree that prices exactly both zero coupon bonds and caps. More specifically, for
every step i, i = 1, ..., log interest rate zi,j + 1 is set according to the algorithm
zi,j + 1 = zi,j − 2 × σ i ×
√
Δ
for
j = 0, 1, ..., i − 1
(11.23)
where the only difference from Equation 11.22 is that now σ i appears on the right-hand
side. Although not necessary, we go one step further and express zi,j +1 directly as a
function only of top interest rate zi,0 , instead of zi,j . By substituting repeatedly zi,j , zi,j −1
on the right-hand side of Equation 11.23 we find
√
zi,j + 1 = zi,0 − 2 × (j + 1) × σ i × Δ
(11.24)
404
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Table 11.13
i=0
The Black, Derman and Toy model
i=1
i=2
σ2
r2 , u u
σ1
r1 , u
=
=
i=3
=
=
?
?
...
?
?
...
r2 , u d = r2 , u u × e−2 ×σ 2 ×
r0 = 2.17%
r1 , d = r1 , u ×
e−2 ×σ 1 ×
√
Δ
√
Δ
...
...
r2 , d d = r2 , u u ×
√
e−4 ×σ 2 × Δ
...
Using directly the actual interest rates ri,j = ez i , j , Table 11.13 shows the methodology to
construct the tree.
Table 11.14 shows the BDT model that prices both the swap-implied zero coupon bonds
and the caps in Table 11.10. The row denoted by σ i reports the estimated step-i volatility
that was necessary to fit the cap prices. The step-i volatility σ i is called forward volatility,
and it is plotted in Figure 11.2 for maturity up to 10 years.6
Definition 11.4 The forward volatility σ i is the level of volatility in step i in the Black,
Derman, and Toy model that matches the cap price with maturity i + 1.
From the figure, it appears clear that there is a relation between the forward and the flat
volatility. Indeed, it is possible to think of the flat volatility as a sort of weighted average
of forward volatility: If the forward volatility of a cap with maturity i + 1 is higher than
the forward volatility of a cap with maturity i, then the implied volatility of the former cap
is also (likely) higher than of the latter.
Once the BDT model is fitted to both the term structure of interest rates and cap prices,
it can be used to value other derivative securities that depend on interest rates. Chapters 12
and 13 contain several examples.
11.4
RISK NEUTRAL TREES FOR FUTURES PRICES
In this section we elaborate on the behavior of futures prices on risk neutral trees. Because
futures markets, such as the Eurodollar futures or the T-bond futures, are very liquid, traders
extract as much information as possible from the behavior of futures prices to build risk
neutral trees. Before considering specific contracts, we have to find out what the general
behavior of a futures price on an interest rate tree is. Let Fi,j (k) denote the futures price
6 In the figure, the forward volatility σ has been shifted forward by one period, to match the timing of the implied
i
volatility. That is, σ i is the forward volatility in step i, which is obtained by pricing a cap that matures at time
i + 1. For consistency with the implied volatility maturity convention, the figure plots the flat volatility by one
quarter forward.
RISK NEUTRAL TREES FOR FUTURES PRICES
Table 11.14
Time =⇒
Period i =⇒
σ i (%) =⇒
node j
0
1
2
3
4
5
6
7
8
9
10
0
0
0.25
1
405
The Black, Derman and Toy Model on November 1, 2004
0.5
2
0.75
3
1
4
1.25
5
1.5
6
1.75
7
2
8
2.25
9
2.5
10
18.77 18.66 27.77 30.19 29.76 30.75 33.98 31.77 31.99 30.28
2.17
2.68
2.22
3.21
2.66
2.21
4.26
3.23
2.44
1.85
5.37
3.97
2.94
2.17
1.60
6.45
4.79
3.56
2.64
1.96
1.46
7.97
5.86
4.31
3.17
2.33
1.71
1.26
10.58 12.02 14.63 16.36
7.53 8.74 10.62 12.08
5.36 6.36 7.72 8.93
3.82 4.63 5.60 6.59
2.72 3.37 4.07 4.87
1.94 2.45 2.95 3.60
1.38 1.79 2.15 2.66
0.98 1.30 1.56 1.96
0.95 1.13 1.45
0.82 1.07
0.79
of a contract maturing at time k at time/node (i, j). If a trader enters into a futures contract
in (i, j), what is the expected one-period profit? The key characteristics of futures, we may
recall, is that they are marked-to-market daily, that is, at the end of every trading day, profits
/ losses accrue to the account of the trader.
Assume that mark-to-market occurs at the same frequency of the time steps on the tree,
for simplicity (or assume that the time step is one day). From the mechanics of futures
markets, then, the profits per period are given by the change in the futures price between
one period to the next. That is, if the interest rate moves up on the tree from ri,j to ri+1,j ,
for instance, then the profit from the futures is N × (Fi+1,j (k) − Fi,j (k)), where N is the
contract size.
Since the interest rate tree is risk neutral, by construction, we find that the risk neutral
expected profit from a position in futures is given by
Risk neutral
expected profit / loss
=
=
E ∗ [Fi+ 1 (k) − Fij (k)]
(11.25)
1
1
× (Fi+ 1,j (k) − Fi,j (k)) + (Fi+1,j +1 (k) − Fi,j (k))
2
2
(11.26)
The key question is the following: If all market participants were risk neutral, what should
the expected risk neutral profit be? Because it costs nothing to enter into a futures position,
the answer is the expected profit should be zero. In fact, if the risk neutral expected profit
was positive, then risk neutral agents, who care only about expected profits and not about
risk, would go infinitely long in the contract, pushing up the futures price. Similarly, if the
risk neutral expected profits from futures was negative, all risk neutral agents would short
the futures.7
7 Recall,
once again, that the risk neutral pricing methodology is not to assume that agents are risk neutral, but
rather that the probabilities along the tree have been changed to incorporate the risk aversion of the market
participants. See again Chapter 9.
406
RISK NEUTRAL TREES AND DERIVATIVE PRICING
The key implication of the risk neutral pricing methodology applied to futures is then
the following restriction:
Risk neutral expected profit / loss = 0 =⇒ E ∗ [Fi+1 (k) − Fij (k)] = 0
(11.27)
Using Equation 11.26, we then obtain a relation between the futures price in node (i, j)
and the two subsequent nodes.
Fi,j (k) =
1
1
× Fi+ 1,j (k) + Fi+1,j +1 (k)
2
2
(11.28)
Equation 11.28 is going to be our main tool to build a risk neutral tree for futures. This
equation allows us to move backward on the tree, exactly as we did for other securities:
given the futures prices at nodes at time i + 1, we can compute the futures price at node i.
One final step in the construction of the risk neutral tree for futures prices is to find the
final value, at maturity, of the futures price. We use the convergence property of futures
prices: At maturity, the futures price must converge to the value of the security underlying
the futures contract. Denote by Vi,j the final payoff from the futures contract at maturity i.
For instance, in a T-bond futures Vi,j is the value of the appropriate Treasury bond scheduled
for delivery in node (i, j). The convergence property then requires that at maturity i, the
futures price equals the value of the security underlying the futures contract
Fk ,j (k) = N × Vk ,j
(11.29)
where k = maturity. From the risk neutral interest rate tree, we can obtain the value of
Vk ,j at maturity k of the futures contract, and thus obtain the final condition for the risk
neutral futures tree. Equation 11.28 then provides the backward methodology to build the
rest of the tree. The next two sections provide two examples, by focusing on the Eurodollar
futures and T-bond futures.
11.4.1
Eurodollar Futures
We discussed the details of Eurodollar futures in Section 6.1 of Chapter 6. Recall from Table
6.3 in that chapter that the underlying final cash payment depends on N ×(3-month LIBOR)
where N = $1, 000, 000 is the contract size. From the same table we remember that
the Eurodollar futures contract with maturity k in node (i, j) is quoted as Fi,j (k) =
(100 − fi,j (k)) where fi,j (k) is the futures rate, in percentage.
Denoting by r4 (i, j) the 3-month LIBOR rate in node (i, j), it follows then that the
Eurodollar futures LIBOR rate at maturity must converge to
fk ,j (k) = N × r4 (k, j)
(11.30)
where k = maturity, and recall the notation rn (i, j) denotes the rate with n compounding
frequency, at time/node (i, j).
Consider again the Black, Derman and Toy model in Table 11.14, which was fitted to
the swap rates and cap prices on November 1, 2004. Assume first for simplicity that the
futures contract also matures at quarters i = 1, 2, 3, and 4 in Table 11.14.8 Recall also
8 Because
Eurodollar futures contract expirations are in the March cycle, these maturities are all off by 1 1/2
months, but it simplifies the discussion to first assume that the futures themselves also mature at the same time of
the swaps used in the fitting exercise.
407
RISK NEUTRAL TREES FOR FUTURES PRICES
Table 11.15
i
j
0
1
j
0
1
2
j
0
1
2
3
j
0
1
2
3
4
3-Month Futures Rates Tree
0
1
2
3
4
2.46
2.69
2.23
6-Month Futures Rates Tree
2.69
2.95
2.44
2.90
3.30
2.50
3.76
2.85
2.15
2.18
2.69
2.23
3.22
2.67
2.21
i
j
0
1
j
0
1
2
3.22
2.67
2.21
9-Month Futures Rates Tree
j
0
1
2
3
4.28
3.24
2.45
1.86
1-Year Futures Rates Tree
4.28
3.24
2.45
1.86
Eurodollar Futures Trees
5.41
3.99
2.95
2.18
1.61
j
0
1
2
3
4
3-Month Eurodollar Futures Price
0
1
2
3
4
97.54
97.31
97.77
6-Month Eurodollar Futures Price
97.31
97.05
97.56
96.78
97.33
97.79
9-Month Eurodollar Futures Price
97.10
96.70
97.50
96.24
97.15
97.85
97.82
97.31
97.77
96.78
97.33
97.79
95.72
96.76
97.55
98.14
1-Year Eurodollar Futures Price
95.72
96.76
97.55
98.14
94.59
96.01
97.05
97.82
98.39
that the Black, Derman, and Toy model generates an interest rate tree in which rates are
continously compounded. The 3-month LIBOR rate underlying the Eurodollar futures
contract is instead linearly compounded, which implies that the futures rate at a given
maturity k must equal
fk ,j (k) = r4 (k, j) = 4 × er k , j ×0.25 − 1
(11.31)
where we assume the contract size is N = 1. Table 11.15 reports the trees for the Eurodollar
futures prices with 3-, 6-, 9- and 12-month maturities. Consider first the left-hand side
trees in Table 11.15: The first tree at the top represents the tree of the futures rate fi,j : The
last two entries at i = 1 are given by Equation 11.31, where recall that r4 (1, j) is obtained
from r1,j in Table 11.14. The root of the tree is obtained from the no arbitrage condition
in Equation 11.28. That is, f0 (1) = f1,0 (1)/2 + f1,1 (1)/2. Similarly, the 6-month futures
rate tree, right below, starts at the end with Equation 11.31 for k = 2, and again rk ,j stems
from Table 11.14. Given the final condition, we move backward to obtain the whole tree
for the futures price, using Equation 11.28.
The right-hand side trees in Table 11.15 simply contain the implied quotes for the
Eurodollar futures, as it is convention to quote it as Fi,j (k) = 100 − fi,j (k).
The methodology above implies that given an interest rate tree, such as the one in Table
11.14, we can obtain the implied Eurodollar futures prices. As we did for the implied
volatilities from cap prices, we can also reverse the methodology and obtain the tree that
is consistent with quoted Eurodollar futures prices. Indeed, Eurodollar futures are much
more liquid instruments than swaps, the securities used to obtain the tree in Table 11.14.
408
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Thus, instead of using swaps, traders typically use Eurodollar futures with maturities up to
3-years (the most liquid ones) to obtain the tree.
11.4.2
T-Note and T-Bond Futures
We introduced T-note and T-bond futures in Chapter 6. In this section we go over some
features of the 10-year Treasury note futures and the 30-year Treasury bond futures, as their
characteristics generate some interesting features that deserve study. Table 6.2 in Chapter 6
describes the term of the 10-year T-note futures contract. Upon examination of the terms of
the futures contract, we realize that the party who is taking a short position in these futures,
and thus commits to deliver the underlying security at maturity, is actually also acquiring
implicitly some valuable options. They are as follows:
1. Quality option: There are several securities that are eligible for delivery: For the
10-year contract, these are all the Treasury notes that have a maturity between 6 1/2
and 10 years. For the 30-year Treasury Bond futures the deliverable securities must
have at least 15 years to maturity (or to call). It follows that the trader who is short
the futures contract has the choice of which Treasury note or bond to deliver. Across
all the securities that are eligible for delivery, the short trader will choose the one that
is least expensive, which is then called cheapest-to-deliver. The cheapest-to-deliver
note or bond is not always the same over the life of the futures, as we will see below
in an example.
2. Wild card option: There is a whole month to deliver the note or bond. During
this month the futures contract trades until the seventh business day before the last
business day. Every trading day in the delivery month, the short may deliver until
8 pm (Chicago time), while the contract stops trading at 2 pm (Chicago time). The
trader who is short the futures contract can then participate in any price decline in
the note or bond prices between 2 pm to 8 pm every day. Essentially, the trader short
the contract has about 15 sequential six-hours put options during each day of the
delivery month until the last day of futures trading.
3. End-of-month option: Trading stops seven days prior to the last business day of the
contract month. However, delivery can occur up to the last trading day. Before the
invoice price of the futures has been fixed on the last trading day, but bond prices
keep trading, the short has a timing option as it can select any day during the last
week to deliver.
The existence of these options has an impact on the futures price itself. Consider for
instance the quality option. The short trader will deliver the bond that is least expensive.
Clearly, if no correction was made to the futures price, the cheapest bond or note would
correspond to the one with the lowest maturity and coupon. To avoid liquidity issues and
manipulations, it is desirable to make all of the securities eligible for delivery as similar
as possible, i.e. to standardize them. The exchange achieves this result by modifying the
(futures) price that the long side of the contract will pay in exchange for the bond or note
that it receives. In particular, the invoice price paid by the long side to the short side equals
the futures price Ft ∗ at maturity t∗ multiplied by a conversion factor C. This conversion
factor is defined to make the deliverable bond comparable to a 6% coupon bond or note.
RISK NEUTRAL TREES FOR FUTURES PRICES
Table 11.16
1.) @
2.)
3.)
4.)
5.)
6.)
7.)
8.)
9.)
10.)
11.)
12.)
13.)
14.)
409
Conversion Factors for 10-Year Treasury Note Futures
Coupon
Issue
Date
Maturity
Date
Cusip
Number
Issuance
(Billions)
3 1/2
4
4 1/8
4 1/4
4 1/4
4 1/4
4 1/2
4 1/2
4 1/2
4 5/8
4 5/8
4 3/4
4 7/8
5 1/8
02/15/08
02/15/05
05/16/05
11/15/04
08/15/05
11/15/07
11/15/05
02/15/06
05/15/07
11/15/06
02/15/07
08/15/07
08/15/06
05/15/06
02/15/18
02/15/15
05/15/15
11/15/14
08/15/15
11/15/17
11/15/15
02/15/16
05/15/17
11/15/16
02/15/17
08/15/17
08/15/16
05/15/16
912828HR4
912828DM9
912828DV9
912828DC1
912828EE6
912828HH6
912828EN6
912828EW6
912828GS3
912828FY1
912828GH7
912828HA1
912828FQ8
912828FF2
$23.0
$23.0
$22.0
$23.0
$21.0
$21.0
$21.0
$21.0
$21.0
$21.0
$21.0
$21.0
$21.0
$21.0
Mar.
2008
Jun.
2008
0.8174
0.8902
0.8941
0.9069
0.8983
0.8747
0.9105
0.9080
0.8968
0.9095
0.9074
0.9122
0.9275
0.9450
0.8210
0.8937
0.8971
—–
0.9012
0.8771
0.9128
0.9105
0.8990
0.9115
0.9095
0.9140
0.9293
0.9463
6% Conversion Factors
Sep.
Dec.
Mar.
2008
2008
2009
0.8244
—–
0.9003
—–
0.9040
0.8797
0.9153
0.9128
0.9013
0.9136
0.9115
0.9158
0.9310
0.9478
0.8281
—–
—–
—–
0.9069
0.8821
0.9177
0.9153
0.9034
0.9157
0.9136
0.9177
0.9328
0.9491
0.8317
—–
—–
—–
—–
0.8848
0.9202
0.9177
0.9058
0.9179
0.9157
0.9195
0.9346
0.9506
Jun.
2009
0.8354
—–
—–
—–
—–
0.8873
—–
0.9202
0.9080
0.9200
0.9179
0.9215
0.9365
0.9519
Footnotes: "@" indicates the most recently auctioned U.S. Treasury security eligible for delivery.
The information contained in this publication is taken from sources believed to be reliable, but is not guaranteed by the
CME Group as to its accuracy or completeness, nor any trading result,
and is intended for purposes of information and education only. The Rules and Regulations of the CME Group should be
consulted as the authoritative source on all current contract
specifications and regulations. To obtain updated conversion factors, please visit the Exchange’s website at www.cmegroup.com.
Source: CBOT web site: http://www.cbot.com/cbot/pub/cont detail/0,3206,1391+20356,00.html accessed on June 11, 2008.
The exact formula determining the conversion factor is given by the price of a security
that is equivalent to the delivered note, but with a constant yield equal to 6%. That is,
setting y = .06, the conversion factor of a bond with a coupon equal to c for a futures with
expiration date t∗ is given by9
n
C = Pc (t∗ , T ) =
i= 1
c/2
(1 + y/2)
2×(T i −t ∗ )
+
1
(1 + y/2)
2×(T n −t ∗ )
(11.32)
For instance, Table 11.16 reports some of the conversion factors to be applied to the
various notes eligible for delivery on March 2008 and following months for the 10-year
Treasury note futures. In this case, all of the notes that are eligible for delivery have
a coupon that is lower than 6%, and thus the futures price will be adjusted downwards
(C < 1) at maturity of the futures contract in order to determine the invoice price.
How does the short trader determine the best bond to deliver? Denote by Fi,j (k) the
futures price with maturity k in time/node (i, j). Let there be n notes that are eligible for
delivery. For each note h, h = 1, . . . , n, let C h denote its conversion factor, and let Pkh,j
denote its clean price. If the short trader elects to deliver a given bond h, he or she will
receive then Fk ,j (k) × C h in exchange of a security with clean price Pkh,j . For each note
h we can compute the difference in price, called the basis:
Basis of Note h = Pkh,j − Fk ,j (k) × C h
n
(11.33)
convenient formula is the following: Let a = 1/(1 + y/2). Then
= (a − a n + 1 )/(1 − a). It
j=1
follows that assuming that (a) there are exactly n coupon payments between the futures maturity and the bond
maturity, and (b) the futures delivery date is on a coupon date, the conversion factor is given by the simpler
formula C = c/2 ∗ (a − a n + 1 )/(1 − a) + a n . If (b) is not satisfied, a small time adjustment needs to be made.
9A
410
RISK NEUTRAL TREES AND DERIVATIVE PRICING
The bond h with the smallest basis is called cheapest-to-deliver. 10 It is important to note
that the basis cannot be negative during the delivery month, otherwise a simple arbitrage
exists. In fact, if this was the case, a trader could short the futures, buy the bond and
immediately deliver it, making an instant profit.
How does the presence of these options affect the futures price? We now build a tree
along the lines of the previous sections but to illustrate the impact of these implicit options
on futures prices. For simplicity, we only consider the quality option in a simplified
environment.
EXAMPLE 11.7
Consider the interest rate tree obtained in Example 11.1 by fitting the Ho-Lee model
to the term structure of interest rates on January 8, 2002. By using a procedure similar
to the one illustrated in that example, the resulting interest tree in Table 11.2 can be
extended to longer horizons to price longer-term notes and bonds. More specifically,
Panel A of Table 11.17 contains the zero coupon bond data on January 8, 2002 up to
maturity T = 8. Panel B contains the fitted Ho-Lee model (the first part is the same
as the one in Table 11.2).
Consider now for instance a 10-year Treasury note futures, with maturity of one
year (December, 2003). According to the terms of the contract, described in Table 6.2
in Chapter 6, only Treasury notes with maturities between 6 1/2 and 10 years can be
delivered. As explained above, the futures price will be adjusted by the appropriate
conversion factor to make each possible deliverable bond comparable to a 6% bond.
To illustrate the tree methodology to T-note futures, it is convenient to first consider
the case in which the security underlying the future is exactly a 6%, 7-year Treasury
note (assuming one exists). Given the risk neutral interest rate tree in Panel B, we
can compute the risk neutral tree for the deliverable T-note, which is contained in
Panel C of Table 11.17. The T-note priced on this tree is a 8-year, 6% note, rather
than 7-year note, the reason being that the T-note must have 7 years to maturity at the
maturity of the futures contract, in one year from the present.
If this was the only possible deliverable security, what would be the corresponding
futures price at maturity? We know that at maturity the futures price must converge
to the value of the underlying security to avoid an arbitrage. In other words, denoting
by k = 2 the node corresponding to the maturity of the futures contract (one year
from now), we must have
Fk ,j (k) = Pk ,j
(11.34)
where Pk ,j is the price of the bond at time k and node j. What about the futures
price before maturity? At this point, we proceed backward by using Equation 11.28.
Table 11.18 shows the risk neutral futures price tree.
Consider now the case in which in addition to the 6% note described earlier, two
addition notes are also available, one with a 3% coupon and one with a 9% coupon.
Assume all of these notes have the same maturity. Using the same interest rate tree
10 Note
that the accrued interest is not part of the basis, as both the futures price and the T-note price are clean
prices. In other words, suppose the short trader has to purchase bond k from the market in order to deliver it.
Then, the short trader would pay P kh, j plus accrued interest, but he/she will receive Fk , j × C h plus accrued
interest. Thus, the basis computed using values instead of clean prices is still given by Equation 11.33.
RISK NEUTRAL TREES FOR FUTURES PRICES
411
as in Panel B of Table 11.17 we can obtain the trees of these T-notes as well. They
are contained in Panel A and Panel B of Table 11.19. The next step is to compute
the conversion factor for each of these notes. We can apply the formula in Equation
11.32 to each of these notes, and obtain
Conversion factor 3% note = C 1
=
0.830558903
(11.35)
Conversion factor 6% note = C
2
=
1
(11.36)
Conversion factor 9% note = C
3
=
1.169441097
(11.37)
Consider now the futures maturity date k. For each node j, we can compute the basis
for each bond. We know that at each of these nodes (k, j), the trader who is short
the futures will choose to deliver the bond with the smallest basis. Node by node, we
therefore compute
Node (k, j) : min Pkh,j − Fk ,j (k) × C h
h
(11.38)
The futures price at time k in node j, Fk ,j (k), will move to prevent arbitrage, so that
for every j the following must occur
Node (k, j) : min Pkh,j − Fk ,j (k) × C h = 0
h
(11.39)
where the minimization is taken across the bonds h = 1, ..., n that are eligible for
delivery. In other words, the futures price moves to make the bond price with the
smallest basis in fact equal to the futures price (corrected by the conversion factor).
That is, Fk ,j (k) is given by
Fk ,j (k) = min
h
Pkh,j
Ch
(11.40)
Once Fk ,j has been computed, the rest of the risk neutral futures tree follows from
Equation 11.28. Table 11.20 illustrate the calculations. The last three columns report
Ph
the converted bond price, namely, Ck h, j for each bond h (3%, 6% and 9%) and for
each interest rate node j = 0, 1, 2. The futures price at each node j will equal the
minimum across each row. That is, the futures price in node (k, j) = (2, 0) is given by
F2,0 (2) = 87.86 which corresponds to the converted note price of the 3% note. This
is the minimum across the three bonds for that particular interest rate, and thus the
cheapest-to-deliver is the 3% T-note. Similarly, F2,2 (2) = 117.28 corresponds to the
converted price of the 9% Treasury note, which in this case is the minimum across all
three available notes. The cheapest-to-deliver is the 9% T-note. This finding implies
that depending on whether interest rates increase or decrease, the T-note that is the
cheapest-to-deliver alternates between T-notes with different coupons. The rest of
the tree, as mentioned, stems from the general Equation 11.28.
The futures prices in the tree in Table 11.20 are always lower than the corresponding futures prices for the case when only the 6% Treasury note was available. This
lower futures price reflects the option that is implicit in the futures contract. The other
two options (wild card option and end-of-the-month option) would also decrease the
futures price.
Table 11.17
The 6% Bond Tree
Panel A. Zero Coupon Bond Data
0
0
1
2
1.5
3
2
4
2.5
5
3
6
3.5
7
4
8
4.5
9
5
10
5.5
11
99.1338 97.8925 96.1462 94.1011 91.7136 89.2258 86.8142 84.5016 82.1848 79.7718 77.4339
6
12
6.5
13
7
14
7.5
15
8
16
75.292
72.961
70.865
68.677
66.764
Panel B. The Fitted Ho-Lee Interest Rate Tree
θ i (×100) 1.5675
node j
0
1.74
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
2.1824
1.4374
1.7324
0.7873
0.0423
-0.0628
0.4322
0.9271
0.1202
-0.5194
1.5300
-0.7335 1.0813 -1.0233 0.7313 -1.7140
3.75
1.30
6.06
3.61
1.17
8.00
5.56
3.11
0.66
10.09
7.65
5.20
2.75
0.31
11.71
9.26
6.82
4.37
1.92
-0.52
12.95
10.51
8.06
5.61
3.17
0.72
-1.73
14.15
11.70
9.25
6.81
4.36
1.91
-0.53
-2.98
15.59
13.14
10.69
8.25
5.80
3.35
0.91
-1.54
-3.99
17.27
14.83
12.38
9.93
7.49
5.04
2.59
0.15
-2.30
-4.75
18.56
16.11
13.66
11.22
8.77
6.32
3.88
1.43
-1.02
-3.46
-5.91
19.52
17.07
14.63
12.18
9.73
7.29
4.84
2.39
-0.05
-2.50
-4.95
-7.39
21.51
19.06
16.61
14.17
11.72
9.27
6.83
4.38
1.94
-0.51
-2.96
-5.40
-7.85
22.36
19.92
17.47
15.02
12.58
10.13
7.68
5.24
2.79
0.35
-2.10
-4.55
-6.99
-9.44
24.13
21.68
19.24
16.79
14.34
11.90
9.45
7.00
4.56
2.11
-0.34
-2.78
-5.23
-7.68
-10.12
24.84
22.39
19.95
17.50
15.05
12.61
10.16
7.71
5.27
2.82
0.37
-2.07
-4.52
-6.97
-9.41
-11.86
26.43
23.98
21.54
19.09
16.64
14.20
11.75
9.30
6.86
4.41
1.96
-0.48
-2.93
-5.38
-7.82
-10.27
-12.72
70.17
75.05
80.28
85.90
91.93
98.41
105.36
112.84
120.86
129.48
138.74
71.88
76.10
80.58
85.33
90.37
95.73
101.41
107.44
113.85
120.65
127.86
135.53
74.48
78.02
81.74
85.64
89.73
94.03
98.53
103.26
108.22
113.42
118.88
124.60
130.61
78.52
81.36
84.30
87.35
90.51
93.79
97.19
100.72
104.37
108.16
112.09
116.17
120.39
124.78
83.79
85.83
87.92
90.06
92.26
94.51
96.82
99.18
101.60
104.08
106.62
109.22
111.89
114.62
117.42
90.97
92.09
93.22
94.37
95.53
96.71
97.90
99.10
100.32
101.56
102.81
104.07
105.35
106.65
107.96
109.29
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
Panel C. The 6%, 8-Year Treasury Note Tree
j
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
106.77
96.83
112.57
88.93
102.40
118.21
82.83
94.50
108.04
123.76
78.11
88.31
100.01
113.45
128.89
74.65
83.65
93.85
105.44
118.61
133.57
72.17
80.14
89.09
99.13
110.41
123.09
137.35
70.46
77.54
85.39
94.11
103.80
114.56
126.52
139.82
69.49
75.77
82.65
90.21
98.52
107.65
117.67
128.69
140.80
69.35
74.90
80.92
87.46
94.56
102.28
110.66
119.76
129.65
140.40
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Data
0.5
1
412
Time T
Period i
IMPLIED TREES: FINAL REMARKS
Table 11.18
Futures Price Tree if only a 7-year, 6% Note is Available for Delivery
Period i
Node j
0
1
2
11.5
413
0
1
2
102.98
95.66
110.30
88.93
102.40
118.21
IMPLIED TREES: FINAL REMARKS
The methodology developed in this section is rather powerful. The Black, Derman and
Toy model, for instance, is able to fit the whole term structure of interest rates and the
option prices. This is remarkable. Moreover, it implies that in principle, on the tree, we
can replicate caps using zero coupon bonds, or a short-term cap using long-term caps, and
so on. However, all this comes with a cost, and the cost is called overfitting. The model
has a number of degrees of freedom equal to the number of securities we want to price.
Therefore, it will lack stability of the parameters (e.g. σ i ’s) and won’t have any predicting
power. Indeed, traders using these models simply refit the model day by day in order to
have a single (big) model able to price all of these interest rate securities at once.
The question is, then, what do we do with this model now? By construction, the model
exactly prices zeros and caps, so we cannot use this model to price those. There are two
answers:
1. The model is useful for computing hedge ratios. If a trader sells a particular cap,
he may want to hedge it by taking a position in the underlying swaps. How can the
trader hedge the exposure? As we discussed in Chapter 9, a binomial tree offers the
answer. The hedge ratio is always given by
Hedge ratio =
c1,u − c1,d
V1,u − V1,d
(11.41)
where c1,u and c1,d are the values of the interest rate security sold (e.g., a cap) in
the two interest rate scenarios r1,u and r1,d , and V1,u and V1,d are the values of the
interest rate security chosen to hedge the exposure (e.g., a swap) in the same nodes.
2. Once fitted to zeros and caps, the model can be used to obtain the price of other
interest rate securities, such as structured notes, swaptions, American swaptions, and
so on. The price of these additional derivative securities will be naturally in line with
those of caps that have been used to price them. In particular, we could hedge the
more complicated security using the basic, and more liquid securities, such as caps.
11.6
SUMMARY
In this chapter we covered the following topics:
0
1
2
3
86.77
79.01
93.04
72.97
85.04
99.26
68.47
78.97
91.21
105.47
4
3% and 6% Treasury Note Price Trees
Panel A. The 3%, 8-Year Treasury Note Tree
5
6
7
8
9
10
11
65.15 62.93 61.57 60.93 60.98 61.82 63.59 66.27
74.38 71.11 68.88 67.46 66.81 67.02 68.20 70.29
85.01 80.44 77.10 74.72 73.23 72.68 73.16 74.56
97.26 91.06 86.35 82.81 80.29 78.83 78.48 79.10
111.39 103.16 96.78 91.82 88.06 85.52 84.20 83.92
116.96 108.53 101.84 96.62 92.79 90.35 89.04
121.77 113.01 106.03 100.71 96.97 94.47
125.45 116.39 109.31 104.07 100.25
127.80 118.68 111.71 106.39
128.87 119.93 112.91
128.76 119.83
127.19
12
13
14
15
16
69.86
73.26
76.84
80.59
84.53
88.67
93.01
97.57
102.35
107.37
112.64
118.17
123.98
74.91
77.66
80.52
83.48
86.54
89.73
93.03
96.45
100.00
103.69
107.51
111.47
115.58
119.84
81.28
83.27
85.32
87.41
89.56
91.76
94.02
96.33
98.70
101.12
103.61
106.16
108.77
111.44
114.18
89.65
90.75
91.87
93.00
94.14
95.30
96.47
97.66
98.86
100.08
101.31
102.56
103.82
105.10
106.39
107.70
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
79.11
82.79
86.64
90.69
94.94
99.39
104.05
108.95
114.08
119.47
125.11
131.04
137.25
82.12
85.05
88.08
91.22
94.48
97.85
101.35
104.98
108.74
112.64
116.68
120.87
125.21
129.72
86.30
88.39
90.52
92.72
94.96
97.26
99.62
102.03
104.50
107.04
109.63
112.29
115.01
117.81
120.67
92.29
93.43
94.58
95.74
96.92
98.12
99.32
100.55
101.78
103.04
104.30
105.59
106.89
108.20
109.54
110.88
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
Panel B. The 9%, 8-Year Treasury Note Tree
Node j
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
126.77 114.66 104.88 97.19 91.07 86.38 82.77
132.10 119.77 110.03 102.24 96.18 91.41
137.16 124.88 115.02 107.27 101.07
142.05 129.64 119.83 111.91
146.39 134.05 124.04
150.18 137.65
152.92
80.00
87.62
96.06
105.41
115.78
127.27
140.03
154.18
78.01
84.72
92.07
100.14
108.98
118.68
129.32
141.00
153.81
76.88
82.78
89.16
96.09
103.61
111.76
120.61
130.20
140.62
151.93
76.75
81.89
87.40
93.31
99.66
106.46
113.76
121.60
130.00
139.03
148.71
77.50
81.92
86.60
91.56
96.83
102.42
108.35
114.63
121.31
128.38
135.89
143.86
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Period i
Node j
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
414
Table 11.19
SUMMARY
415
Table 11.20 Treasury Note Futures when 3%, 6% and 8% Notes Are Available
Period i
Node j
0
1
2
0
1
2
102.48
95.12
109.83
87.86
102.38
117.28
Converted Note Prices Pik∗ , j /C k
3% Note 6% Note 9% Note
87.86
88.93
89.68
102.38
102.40
102.42
119.51
118.21
117.28
1. Risk neutral trees: Risk neutral trees are binomial trees that are specifically designed to price interest rate securities, as the probabalities of upward and downward
movements are risk neutral. Three examples are:
(a) Ho-Lee model. The main variable is the level of the short-term interest rates.
The risk neutral tree is set up to exactly fit the term structure of interest rates.
With sufficient steps, the probabality distribution of interest rates in the future
converges to a bell-shaped normal distribution. One drawback is that it allows
for negative interest rates.
(b) Simple Black, Derman, and Toy (BDT). In the simple BDT model the main
variable is the natural logarithm of the short-term interest rate. Like the Ho-Lee,
the simple BDT model also fits exactly the term structure of interes rates, but it
does not allow for negative interest rates. With sufficient steps, the probability
distribution of interest rates in the future converges to a log-normal distribution.
One drawback is that it gives too little probability to low interest rates.
(c) Black, Derman, and Toy. Unlike the Ho-Lee and the simple BDT, the full BDT
model is also able to fit the term structure of volatility of caps, as it assumes
that the volatility of interest rates can be different step after step. The simple
BDT model is obtained as a special case by assuming constant volatility.
2. Caps and floors: These are portfolios of options, called caplets or floolets, that pay
when the short-term interest rate is above a strike rate (caps) or below a strike rate
(floors). They can be priced on a binomial tree, through the construction first of a
cash flow tree, which defines the cash flow that is paid by the cap or floor along the
tree.
3. Empirical interest rate volatility: This is the volatility of interest rate movements
computed from the historic time series of interest rate changes. For the Ho-Lee model,
such volatility can be computed as the annualized standard deviation of interest rate
changes, while for the simple BDT model such volatility can be computed as the
annualized standard deviation of changes in log interest rates.
4. Cap/floor implied volatility. The volatility of the interest rate process that matches
exactly a given cap or floor is called the cap/floor implied volatility. It is also called
flat volatility.
5. Cap/floor forward volatility. This is the volatility that prices a given caplet/floorlet
in a cap/floor. The implied (flat) volatility can be thought of as a weighted average
416
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Table 11.21
i=0
An Interest Rate Tree
i=1
i=2
r2 , u u = 0.1299
r1 , u = 0.0868
r0 = 0.04
r2 , u d = 0.0723
r1 , d = 0.0268
r2 , d d = 0.0147
of forward volatilities. The forward volatility can be computed while fitting the full
BDT model to bonds and caps.
6. Swaps and swaptions: Swaps are contracts to exchange cash flows in the future,
according to a specific formula. Fixed-for-floating swaps, the most common, entail
a counterparty to pay a fixed coupon in exchange for a floating rate that depends on
a reference rate. Swaptions are options to enter into a given swap at some time in
the future. In a receiver swaption, the option holder has the right to enter into a swap
as a fixed rate receiver, while in a payer swaption the option holder has the right to
enter into a swap as a fixed rate payer.
7. Risk neutral binomial trees for futures: Futures contracts are marked-to-market. The
profits and losses within a short period are just given by the change in the futures
price. The risk neutral methodology implies that the futures price today is equal to
the risk neutral expected futures price tomorrow. Given the convergence property of
futures prices, it is possible to construct the futures tree by moving backward from
the futures price at maturity, which equals the underlying value.
8. Cheapest-to-deliver: In T-bond and T-note futures, the short side has the option to
deliver any note or bond within a class of deliverable Treasury securities. Although
an adjustment is made to adjust the futures price to the coupon and maturity of the
bond, still there is a bond that is the least expensive to deliver at maturity. This bond
is called the cheapest-to-deliver. Which bond has this characteristics depends on the
interest rate itself.
11.7
EXERCISES
1. You have estimated the risk neutral model for the continuously compounded interest
rate as in Table 11.21. There is equal probability to move up or down the tree and
each interval time represents 1 year, that is, Δ = 1.
(a) Compute the current (i = 0) zero coupon spot curve, for all possible maturities;
EXERCISES
Table 11.22
i=0
417
An Interest Rate Tree
i=1
i=2
r2 , u u = 0.04
r1 , u = 0.03
r2 , u d = 0.025
r0 = 0.02
r2 , d u = 0.019
r1 , d = 0.015
r2 , d d = 0.01
(b) Compute the price of a security that pays $100 at time i = 2 if the interest rate
at that time is .0723 and zero otherwise. That is,
100 if r (2) = .0723
CF (2) =
0
otherwise
(c) Compute the price of a 3-year floor with strike rate rK = .04 and notional 100.
Recall that a floor pays at time i + 1 the cash flows (determined at time i):
CF (i + 1) = N × max (rK − r1 (i) , 0)
where r1 (i) is the annually compounded rate at time i.
(d) Compute the spread between the 2-year zero coupon yield and the 1-year zero
coupon yield, both at time i = 0 and at time i = 1. What is the expected
change in the spread?
(e) Suppose that you want to hedge against an increase in the spread between i = 0
and i = 1. What is the price of a European spread call option, with 1 year to
maturity, strike spread sp = .008 and notional amount N = 100? The spread
call option pays
CF (1) = N × max (Spread (1) − sp, 0)
where “Spread (1)” is the spread between the 2-year and the 1-year continuously compounded zero coupon yields as of time i = 1 (that is, 1 year from
now).
2. You have estimated the model for continuously compounded interest rates in Table
11.22. There is equal risk neutral probability to move up or down the tree and each
interval time represents 1 year, that is, Δ = 1.
(a) Compute the current (i = 0) zero-coupon spot curve, for all possible maturities.
418
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Table 11.23
i=0
An Interest Rate Tree
i=1
i=2
r2 , u u = 0.1
r1 , u = 0.07
r2 , u d
= 0.05
r2 , d u
r0 = 0.04
r1 , d = 0.03
r2 , d d = 0.02
(b) Compute the price of a 3-year, 3% coupon bond.
(c) Compute the price of a standard 3-year floating rate bond, with coupon equal
to the risk-free rate (you should remember the timing of cashflows, here). Is
the price what you expected?
(d) Compute the value of a swap, from the perspective of a fixed rate payer, with
swap rate equal to 3% and notional 100.
(e) Compute the price of a 3-year cap with strike rate 3%, and notional 100. What
do you think the price of a floor, with strike rate 3%, should be?
3. Suppose that you estimated the risk neutral tree for interest rates in Table 11.23,
where there is equal risk neutral probability to move up or down the tree. Assume
also for simplicity that each interval of time represents 1 year, that is, Δ = 1.
(a) Compute the price of a 1-, 2-, and 3-year zero coupon bond.
(b) Compute the swap rate c (3) for a plain vanilla swap with annual cash flows
and maturing on i = 3. Recall that cash flows are given by
(11.42)
CFi,j (i + 1) = N × (r1 (i, j) − c(3))
where, r1 (i, j) = er i , j ×1 − 1 is the annually compounded rate that corresponds to ri,j .
(c) Consider an option with maturity i = 1 with the following payoff
r1 (1, j)
s1 = max 0.98 ×
− Z1 (3) , 0
.06
(11.43)
where r1 (1, j) is the annually compounded rate at time i = 1, and Z1 (3) is the
zero coupon at time i = 1 that pays 1 at time i = 3.
i. What is the value of this option?
EXERCISES
Table 11.24
i=0
419
An Interest Rate Tree
i=1
i=2
r2 , u u = 0.08
r1 , u = 0.06
r2 , u d
= 0.05
r2 , d u
r0 = 0.04
r1 , d = 0.03
r2 , d d = 0.02
ii. If you sell this option, how can you hedge it? Write down the hedging
strategy and confirm its perfomance at i = 1. Be precise in the description
of the steps.
(d) Procter & Gamble Leveraged Swap: In November 1993, Procter & Gamble
(P&G) entered a swap with Bankers Trust (BT) where BT would pay P&G
a fixed rate r, and P&G would pay BT a floating rate plus a spread. The
spread was going to be equal to 0 at time of initiation, and would be set at time
i = 1 equal to the value s1 in Equation 11.43. The spread remains constant
thereafter. To provide an example, suppose that the interest rate increases at
i = 1 to r1,u and decreases afterwards to r2,u d . The spread is set to s1,u at
time i = 1, implying that P&G has to pay at time i = 1 simply r0 × N , at time
i = 2 the cash flow (r1 (1, u) + s1,u ) × N , and at time i = 3 the cash flow
(r1 (2, ud) + s1,u ) × N , where N = 100 is the notional. (Remember that in
swaps, the floating rate at time i determines the cash flows at time i + 1).
i. Assume the maturity of the levereged swap is 3 years. What is the value
of the swap for P&G if r = c (3), where c (3) is the swap rate determined
in Part (b)?
ii. Given your answer to part i, the value of r that makes the swap value equal
0 at i = 0 is higher or lower than c (3)? Provide an brief intuition.
iii. Using a spreadsheet, compute the value of r (this can be done by using
solver in Microsoft Excel, or simply by trial and error).
4. Suppose that you estimated the risk neutral tree for the continuously compounded
interest rates in Table 11.24, where there is equal probability to move up or down the
tree. Assume also for simplicity that each interval of time represents 1 year, that is,
Δ = 1.
(a) Compute the term structure of interest rates at time i = 0.
420
RISK NEUTRAL TREES AND DERIVATIVE PRICING
(b) From the term structure, compute the swap rate c(3) for an annual swap maturing
at i = 3. Use the tree to check that the value of the swap rate is indeed zero.
Take the perspective of the fixed rate payer into your calculations.
(c) An index amortizing swap is a swap whose notional value decreases over time
depending on the interest rate scenario. Consider the index amortizing swap
with initial notional N0 = 100 and with the following characteristics:
• Maturity i = 3
• Ammortization schedule:
if ri
if ri
if ri
if ri
=
=
=
>
.02
.03
.05
.05
then
then
then
then
100 % reduction in notional
50 % reduction in notional
20 % reduction in notional
no reduction in notional
• At i = 0 no ammortization takes place (lockout period).
For example, if at time i = 1 the interest rate declined to r1,d = 0.03, then the
notional to apply to the next payment (at i = 2) is not N = 100 but N = 50. If
instead at i = 1 the interest rate was r1,u = .06, then no reduction in notional
would take place.
i. Intuitively, is the fixed rate payer better off or worse off compared to the
plain vanilla swap in Part (b) if the swap rate of the index ammortizing
swap is also c(3) as computed earlier? Explain. (Hint: Thera are no
computations needed here. Only intuition.)
ii. Obtain the value of the swap for the fixed rate payer assuming c(3) as in
Part (b). (Hint: Recall that the notional becomes path dependent. How
does this fact affect the tree for cash flows?)
iii. Use a spreadsheet to obtain the value of the fixed swap rate for the index
amortizing swap (you can use either a solver, as in Microsoft Excel, or
trial and error). Is this fixed rate higher or lower than c(3) computed in
Part (b)? Discuss.
iv. Suppose you “sold” the swap whose value you found in Part iii. How
would you hedge it? Provide the initial hedge ratio at i = 0. You can
consider any instrument that you like, such as zero coupon bonds or plain
vanilla swaps.
5. Let today be November 3, 2008.
(a) Use the LIBOR rate and the swap data on November 3, 2008 in Table 11.26
and fit the LIBOR curve.
(b) From the LIBOR discount curve, fit the Ho-Lee model of the interest rates,
with quarterly steps. You can use the LIBOR volatility reported in the text, or
estimate the LIBOR volatility yourself. Data on LIBOR are available on the
British Bankers Association Web site (www.baa.org.uk).
(c) Compare risk neutral expected future interest rates to the continuously compounded interest rates. How does the difference depend on the assumed volatility of the interest rate? (Hint: For each assumed volatility of the interest rate,
EXERCISES
Table 11.25
Issuer
Rating
Pricing Date
Maturity Date
Principal
Coupon Frequency
Coupon
Corridor Lower Bound
Corridor Upper Bound
Reference Rate
421
Term Sheet 5-year Corridor Note
HAL Corp.
AAA
November 3, 2008
November 3, 2013
100
Quarterly
5.4% if reference rate within corridor bounds; 0% otherwise
1%
5%
3 Month LIBOR on previous fixing date
you need to refit the tree to make sure that the tree correctly reflects the forward
rates. Do the exercise for 3 values of volatility).
(d) Compute the value of 1-year, 2-year and 3-year cap. Compare your value with
the one in the data, in Table 11.26.
(e) Compute the value of a 5-year swap (the swap rate in Table 11.26) with quarterly
payments (i.e., assume that both floating and fixed payers pay at quarterly
frequency). Is the value of the swap obtained from the tree what you would
expect from first principles?
(f) Use the swap tree computed in Part (e) to compute the value of 1-year, at-themoney swaption to enter into a 4-year swap.
6. On November 3, 2008, the AAA rated company HAL issued a 5 year, corridor note
with a quarterly coupon, according to the term sheet in Table 11.25. Given the
Ho-Lee model fitted in Exercise 5, compute the following:
(a) Obtain the value of the corridor note discussed in Table 11.25.
(b) Compute the value of a straight quarterly coupon bond, and determine the
(annualized) coupon rate that generates a price similar to the one of the corridor
note. Is this coupon higher or lower than 5%? Comment.
(c) Compute the spot rate duration of the corridor note and of the straight bond
obtained in Part (b). Which one is higher? Why?
(d) Consider the future time t = 2 (step i = 8). Plot the value of the corridor
note at i = 8 against the interest rate scenarios at the same time. On the same
graph also report the values of the straight fixed coupon bond. Comment on
the difference between the two securities. Which one appears to have negative
convexity? Why?
(e) Consider the simple BDT model. Fit this model to the same data [again, either
use the LIBOR volatility in the text, or estimate it using the same data as in
Part (b)] and recompute the value of the corridor note. Is this the same value
you obtained using the Ho-Lee model? Comment.
422
RISK NEUTRAL TREES AND DERIVATIVE PRICING
Table 11.26
3 Month LIBOR (%)
Swap Rates and Cap Prices on November 3, 2008
2.8588
Maturity
Swap Rate (%)
Cap Price (×100)
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
3.25
3.50
3.75
4.00
4.25
4.50
4.75
5.00
2.6486
2.4929
2.4320
2.4491
2.4938
2.5561
2.6260
2.7252
2.8630
3.0108
3.1400
3.2471
3.3474
3.4408
3.5270
3.6076
3.6835
3.7531
3.8150
0.0528
0.1313
0.2401
0.3826
0.5405
0.7106
0.8932
1.1095
1.3729
1.6636
1.9502
2.2235
2.4973
2.7711
3.0451
3.3208
3.5968
3.87
4.137
Original Data Source: Bloomberg.
Reported Data are interpolated from quoted swap rates and flat volatilities, and then computed
using Black’s model.
7. Today is November 3, 2008, and the 3-month LIBOR, swap rates and cap prices are
as in Table 11.26.
(a) Fit the LIBOR curve (see Exercise 5).
(b) Compute the implied volatilities from the simple BDT model.
(c) Compute the forward volatilities from the BDT model.
(d) On this tree, value the corridor note discussed in Table 11.25 (see Exercise 6).
CHAPTER 12
AMERICAN OPTIONS
Numerous interest rate securities have embedded American options, that is, the right of
the option holder to receive a payoff of some kind any time before a predefined maturity.
Some of these options are explicitly outlined within the terms of a contract, but others are
somewhat hidden and are only embedded within another interest rate security. Let’s recall
first the definition of an American option (see also Chapter 6).
Definition 12.1 An American call option is a contract between two counterparties in
which one party (the option buyer) has the right, but not the obligation, to buy a given
security at a predetermined price on or before a maturity time T , and the other party (the
option seller) has the obligation to sell such security. An American put option is a similar
contract in which the option buyer has the right to sell a given security at a predetermined
price on or before a maturity time T , and the option seller has the obligation to purchase
such security.
In this chapter we show that the tree methodology developed in Chapters 10 and 11
can be readily adapted to obtain pricing and hedging strategies for this more complicated
security. We apply this methodology to classic American option securities, namely, callable
bonds, American swaptions and mortgage backed securities.
423
424
AMERICAN OPTIONS
Table 12.1
U.S. Treasury Callable Bonds on January 8, 2002
Maturity
Coupon
First Call Date
Bid
Ask
20070215
20071115
20080815
20081115
20090515
20091115
20100215
20100515
20101115
20110515
20111115
20121115
20130815
20140515
20140815
20141115
7.625
7.875
8.375
8.750
9.125
10.375
11.750
10.000
12.750
13.875
14.000
10.375
12.000
13.250
12.500
11.750
20020215
20021115
20030815
20031115
20040515
20041115
20050215
20050515
20051115
20060515
20061115
20071115
20080815
20090515
20090815
20091115
100.5625
104.7188
108.4219
110.1406
112.3438
117.3906
122.4688
118.0000
129.9063
137.2500
140.8750
127.8594
139.2344
150.1406
146.4688
142.4844
100.5938
104.7500
108.4531
110.1719
112.3750
117.4219
122.5000
118.0313
129.9375
137.2813
140.9063
127.9219
139.2969
150.2031
146.5313
142.5469
Source: Data excerpted from CRSP (Daily Treasuries) ©2009 Center for Research in
Security Prices (CRSP), The University of Chicago Booth School of Business.
12.1
CALLABLE BONDS
Many bond issuers, including the U.S. government in the late 1970s and early 1980s, may
issue callable bonds, that is, standard fixed coupon bonds, but ones in which the issuer
retains the option to buy back the bond for its par value during a defined time interval
before maturity. For example, Table 12.1 shows the available U.S. Treasury callable bonds
on January 8, 2002.1
In this table, the November 2012 U.S. Treasury bond is callable at par starting on
November 2007. This implies that after November 2007, the U.S. government can purchase
this bond for $100 any time it desires to do so. (If the U.S. government calls the bond
between coupon days, it has to pay the accrued interest to the bond holder.) Of course, the
U.S. government, or the issuer of a callable bond more generally, will exercise its option to
purchase back a bond only when it is convenient to do so. For instance, the U.S. government
issued numerous callable bonds during the high interest rate period in the early 1980s, with
the expectation that inflation and thus interest rates would decline in the future. If interest
rates effectively decline in the future, it would be beneficial for the U.S. government to
refinance its old bonds bearing a high coupon rate and exchange them for other bonds that
reflect the current lower rates.
How can we determine the optimal timing at which the issuers of callable bonds should
exercise their option to purchase the bonds? How does this optionality affect the pricing
and the hedging of these bonds? The interest rate tree methodology developed in Chapters
10 and 11 provides us a convenient methodology, illustrated in the following example.
1I
will use this date in Example 12.2 to provide a concrete example of the binomial tree methodology. This is the
same date of most examples in Chapter 11
CALLABLE BONDS
425
EXAMPLE 12.1
Consider the 1.5-year, 3% coupon bond we discussed in Section 11.2.1 of Chapter
11, Table 11.4, but assume it is callable at par (100) starting on i = 1 (that is, it
cannot be called at time i = 0). To find the price of the callable bond, we think of
this security as a portfolio composed by two securities: The first security is a straight
non-callable bond, whose value is Pi,j (3) in node (i, j) is computed in Table 11.4
in Chapter 11 and reported in Panel A of Table 12.2. The second security is an
American call option in which, upon calling the bond, the government receives the
payoff
Payoff from call option at node (i, j) = max(Pi,j (3) − 100, 0)
(12.1)
That is, by exercising the option, the government has to pay $100 to the bond holders,
but it receives back the bond with value Pi,j (3). The question is then when it is
optimal to exercise this option.
Moving backward on the tree we can resolve this problem. In fact, at any node
(i, j) the issuer can decide whether to exercise the option or wait. If it exercises, the
payoff (= value of the option) is
Ex
= Pi,j (3) − 100
Ci,j
If it waits, the value of the option
Wait
Ci,j
e−r i , j Δ E ∗ [Ci+1 ]
1
1
= e−r i , j Δ
Ci+1,j + Ci+1,j +1
2
2
=
The option holder should act to maximize the value of its option. Therefore, at
any node (i, j) the option holder will choose between Wait and Exercise so as to
maximize the payoff. It follows that the value of the option at time/node (i, j) is
Wait Ex (12.2)
, Ci,j
Ci,j = max Ci,j
−r i , j ×Δ ∗
E [Ci+1 ] , Pi,j (3) − 100
= max e
Since the option expires worthless at maturity, as the issuer has to redeem the bond
at par value, at maturity I = T /Δ we have
CI ,j = 0 for all j
Given this final value of the option, we can start the backward procedure in Equation
12.2. The resulting tree is in Table 12.2, and the current value of the option to call
the bond early is C0 = 0.1874.
What then is the price of the callable bond? Because the buyer of a callable bond
is really long a non-callable bond and short the American call option (sold to the
issuer), the price of the callable bond is
V0cb (3)
= P0 (3) − C0
=
100.5438 − 0.1874
=
100.3564
426
AMERICAN OPTIONS
Table 12.2 The Call Option in a Callable Bond
t=0
i=0
t = 0.5
i=1
t=1
i=2
t = 1.5
i=3
Panel A: The 3%, 1.5 year non-callable bond
P 3 , u u u (3) = 100
P 2 , u u (3) = 99.1094
P 3 , u u d (3)
= 100
P 3 , u d u (3)
P 1 , u (3) = 99.4667
P 0 (3) = 100.5438
P 2 , u d (3) = 99.7287
P 3 , u d d (3)
= 100
P 3 , d d u (3)
P 1 , d (3) = 100.3780
P 2 , d d (3) = 100.1886
P 3 , d d d (3) = 100
Panel B: The Option to Call
C 3 , u u u (3) = 0
C2 , u u =
= max (99.1094 − 100, 0)
=0
C1 , u =
= max(99.4667 − 100,
e−2 . 9 0 % / 2 × 0)
=0
C0 =
= e−1 . 7 5 % / 2
× 12 (0 + 0.3780)
= 0.1874
C3 , u u d
=0
C3 , u d u
C2 , u d =
= max(99.7287 − 100, 0)
=0
C1 , d =
= max(100.3780 − 100,
e−2 . 1 4 % / 2 (0 + 0.1886)/2)
= 0.3780
C3 , u d d
=0
C3 , d d u
C2 , d d
= max(100.1886 − 100, 0)
= 0.1886
C3 , d d d = 0
CALLABLE BONDS
427
In the example above the issuer calls the bond at time i = 1 if the interest rate moves to
r1,d or r2,dd , but not otherwise. The effect is intuitive, as when the interest rate decreases,
the price of the bond increases, and it becomes profitable for the government to purchase a
highly priced security for only $100. Another way to say this is that the government can
refinance its debt at a cheaper rate when interest rates decline.2
12.1.1
An Application to U.S. Treasury Bonds
Does this “optimal” exercise model work? Is it producing reasonable prices? To check
whether the model described in the previous section is accurate, we can proceed as follows.
We can first use the Simple Black, Derman, and Toy (BDT) model in Section 11.1.2 in
Chapter 11 and fit it to the non-callable bonds. Then, if the methodology described in the
previous subsection for the pricing of American call options is accurate, the model should
then deliver the correct prices for the callable bonds on the same day.The next example
carries out the calculation.3
EXAMPLE 12.2
Table 11.3 in Chapter 11 contains the Simple BDT model fitted to the zero coupon
bonds on January 8, 2002. We can use this tree to price callable bonds. Consider
for instance the callable bond expiring on August 15, 2014 in Table 12.1. The first
call date is August 15, 2009. To price this bond, we need to extend the interest
rate tree in Table 11.3 to a longer maturity. The resulting tree for the American call
option embedded in the Treasury Bond is provided in Table 12.3. In this table, the
shaded areas correspond to the time / node combination in which the exercise of the
American call option takes place. Clearly, before time T = 7.5, the option cannot
be exercised. When T = 7.5, the option is exercised when the interest rate is below
r15,6 = 8.11%.
What is the price of the callable bond then? From Panel A in Table 12.3 the price
of the non-callable bond is equal to $169.732, while from Panel B the value of the
call option is $22.33. Thus, the value of the callable bond is equal to
V0cb (T ) = $169.732 − $22.33 = $147.40
This value is close to the traded value in Table 12.1, which equals $146.5 (average
bid and ask).
As a final comment, note that in the simple three-period example in Table 12.2 it is
always the case that the issuer exercises whenever the option is in the money, that is when
Pi,j − 100 > 0. But this is not always the case, as there is a value to waiting. There are
situations in which an immediate exercise would give a positive payoff, while it may be
optimal to wait anyway, as the option can get even more profitable. This is apparent in
Table 12.3. Consider position (i, j) = (15, 5), in which case the value of the American call
2 In
the example, the interest rate does not actually decline, but it moves below the risk neutral expected interest
rate, pushing up bond prices.
3 The next example carries out the exercise in a simple manner, and abstracting from a few institutional details that
are bound to affect the price. For instance, the U.S. Treasury must announce four months in advance its intention
to call back a bond on the next call date.
428
AMERICAN OPTIONS
Figure 12.1 The Negative Convexity in Callable Bonds
115
One Year to Call
Six Months to Call
Call Date
110
105
Bond Price
100
95
90
85
80
75
70
0
2
4
6
8
10
12
Interest Rate (%)
14
16
18
20
is Ci,j = $8.37. The non-callable bond price in such node is Pi,j (3) = $107.404. Thus,
an immediate exercise would yield Pi,j (3) − 100 = 107.404 − 100 = $7.404. Yet, waiting
yields a higher (risk neutral) expected profit. In fact, by waiting, there is (risk neutral) 0.5
probability of getting $4.59 if interest rates move to ri+1,j and 0.5 probability of getting
$13.10, if interest rates move to ri+ 1,j + 1 . The discounted risk neutral expectation of these
payoffs at time i + 1 is e−10.99% /2 × (4.59 + 13.10)/2 = $8.372, which is higher than the
immediate payoff. Thus, it is beneficial for the issuer (Treasury) to wait in this case.4
12.1.2
The Negative Convexity in Callable Bonds
How does the short American option implicit in a callable bond affect the relation between
its price and interest rates? As we know from basic principles, and especially Chapter
4, there is a natural positive convex relation between interest rates and the price of the
non-callable securities. This is no longer true for callable bonds. Indeed, Figure 12.1
shows the profile of the price of the August 15, 2014 callable Treasury Bond in Example
12.2 around the first call date (August 15, 2009). We see that the bond price convexity is
negative for interest rates below 14%. That is, while lower interest rates still increase the
price of the bond, they do so at an increasingly lower rate, as opposed to an increasing rate
as for any non callable bond. The reason for this behavior is as follows: As interest rates
decline to below 14% it becomes increasingly likely that the Treasury will call back the
bond whenever possible. As a consequence, the bond price has to converge to the call price
($100) as interest rates decline.
4 Recall
from Chapter 9 that a high risk neutral expected cash flow means that there is a replicating portfolio that
yields a higher cash flows than exercising the option does.
Table 12.3 The American Call Tree
Panel A: The non-callable bond
CALLABLE BONDS
time
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
i
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
15.0
16.0
17.0
18.0
19.0
20.0
21.0
22.0
23.0
24.0
25.0
0 169.732 155.631 141.906 129.058 116.933 105.887 95.589 85.705 76.203 67.424 59.768 52.782 45.994 40.868 35.854 32.319 28.701 26.430 23.363 21.038 19.598 19.393 21.289 27.610 45.419 100.000
1
174.299 161.400 149.100 137.286 126.284 115.828 105.636 95.689 86.318 77.927 70.122 62.474 56.452 50.554 46.189 41.778 38.823 35.092 32.276 30.580 30.427 32.737 39.657 56.714 100.000
2
178.451 166.933 155.724 145.103 134.859 124.758 114.788 105.246 96.516 88.264 80.115 73.466 66.930 61.883 56.818 53.226 48.915 45.614 43.568 43.215 45.414 51.945 66.826 100.000
3
182.141 171.669 161.615 151.811 142.065 132.368 122.976 114.228 105.853 97.532 90.527 83.614 78.068 72.520 68.371 63.586 59.819 57.321 56.506 58.102 63.486 75.435 100.000
4
184.957 175.531 166.265 157.007 147.747 138.696 130.144 121.873 113.621 106.476 99.403 93.523 87.652 83.037 77.882 73.679 70.664 69.196 69.861 73.673 82.499 100.000
5
186.892 178.172 169.433 160.666 152.041 143.794 135.755 127.713 120.574 113.488 107.404 101.333 96.341 90.904 86.307 82.757 80.552 80.144 82.261 88.138 100.000
6
187.723 179.472 171.184 162.991 155.082 147.325 139.553 132.500 125.487 119.287 113.103 107.811 102.164 97.224 93.168 90.231 88.755 89.256 92.550 100.000
7
187.404 179.542 171.744 164.158 156.683 149.188 142.255 135.351 129.094 122.850 117.325 111.520 106.293 101.785 98.182 95.736 94.811 95.950 100.000
8
186.063 178.604 171.302 164.081 156.840 150.032 143.245 136.962 130.692 124.987 119.066 113.604 108.714 104.537 101.260 99.140 98.541 100.000
9
183.899 176.835 169.829 162.807 156.113 149.435 143.144 136.864 131.022 125.015 119.367 114.164 109.513 105.554 102.467 100.500 100.000
10
181.068 174.238 167.394 160.799 154.214 147.925 141.644 135.699 129.628 123.834 118.381 113.352 108.847 104.997 101.972 100.000
11
177.588 170.886 164.370 157.863 151.579 145.301 139.278 133.160 127.253 121.606 116.280 111.349 106.907 103.073 100.000
12
173.522 167.071 160.625 154.347 148.073 141.992 135.840 129.846 124.049 118.494 113.236 108.341 103.894 100.000
13
169.100 162.702 156.430 150.161 144.036 137.858 131.800 125.889 120.159 114.650 109.413 104.505 100.000
14
164.257 157.990 151.725 145.568 139.371 133.264 127.266 121.404 115.707 110.211 104.958 100.000
15
159.154 152.893 146.711 140.501 134.356 128.294 122.333 116.495 110.805 105.294 100.000
16
153.762 147.563 141.342 135.170 129.060 123.024 117.080 111.245 105.543 100.000
17
148.195 141.967 135.775 129.628 123.537 117.514 111.572 105.727 100.000
18
142.430 136.223 130.050 123.918 117.836 111.814 105.864 100.000
19
136.556 130.362 124.200 118.074 111.993 105.965 100.000
20
130.594 124.408 118.250 112.125 106.039 100.000
21
124.563 118.381 112.223 106.094 100.000
22
118.477 112.295 106.135 100.000
23
112.349 106.165 100.000
24
106.187 100.000
25
100.000
429
430
Panel B: The option to call
0.0
0.0
22.33
0.5
1.0
19.00
26.06
1.0
2.0
15.77
22.79
29.89
1.5
3.0
12.75
19.56
26.82
33.73
2.0
4.0
9.95
16.40
23.69
30.92
37.45
2.5
5.0
7.46
13.36
20.54
28.02
34.96
40.95
3.0
6.0
5.29
10.48
17.37
24.99
32.32
38.77
44.13
3.5
7.0
3.48
7.80
14.18
21.80
29.48
36.41
42.23
46.96
4.0
8.0
2.08
5.41
11.04
18.46
26.43
33.83
40.16
45.32
49.42
4.5
9.0
1.11
3.44
8.10
15.06
23.18
31.06
37.91
43.54
48.04
51.56
5.0
10.0
0.52
1.96
5.49
11.69
19.77
28.10
35.51
41.65
46.57
50.44
53.42
5.5
11.0
0.20
0.97
3.33
8.42
16.16
24.88
32.89
39.58
44.97
49.21
52.49
55.00
6.0
12.0
0.06
0.40
1.73
5.43
12.37
21.30
29.97
37.28
43.19
47.84
51.45
54.22
56.31
6.5
13.0
0.02
0.14
0.78
3.04
8.62
17.45
26.84
34.85
41.32
46.42
50.38
53.41
55.71
57.44
7.0
14.0
0.00
0.04
0.28
1.45
5.12
13.12
23.27
32.10
39.22
44.83
49.18
52.51
55.04
56.94
58.37
7.5
15.0
0.00
0.01
0.08
0.56
2.63
8.37
19.29
29.09
36.96
43.14
47.93
51.58
54.35
56.43
57.99
59.15
8.0
16.0
0.00
0.00
0.02
0.17
1.07
4.59
13.10
22.85
30.69
36.86
41.64
45.30
48.07
50.16
51.73
52.89
53.76
8.5
17.0
0.00
0.00
0.00
0.04
0.35
2.00
7.81
17.32
24.99
31.02
35.70
39.28
41.99
44.04
45.57
46.71
47.56
48.20
9.0
18.0
0.00
0.00
0.00
0.00
0.08
0.68
3.59
11.52
19.07
25.02
29.63
33.16
35.84
37.86
39.37
40.50
41.34
41.97
42.43
9.5
19.0
0.00
0.00
0.00
0.00
0.01
0.16
1.30
6.29
13.60
19.37
23.83
27.25
29.85
31.80
33.26
34.36
35.17
35.77
36.22
36.56
10.0
20.0
0.00
0.00
0.00
0.00
0.00
0.02
0.32
2.45
8.71
14.16
18.38
21.61
24.05
25.89
27.27
28.29
29.06
29.63
30.05
30.36
30.59
10.5
21.0
0.00
0.00
0.00
0.00
0.00
0.00
0.05
0.65
4.54
9.51
13.35
16.28
18.49
20.16
21.40
22.33
23.02
23.54
23.92
24.20
24.41
24.56
11.0
22.0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.11
1.28
5.55
8.85
11.35
13.24
14.65
15.71
16.49
17.08
17.51
17.84
18.07
18.25
18.38
18.48
11.5
23.0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.23
2.47
5.00
6.91
8.34
9.41
10.21
10.80
11.25
11.57
11.81
11.99
12.12
12.22
12.30
12.35
12.0
24.0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.50
1.97
3.07
3.89
4.50
4.96
5.29
5.54
5.73
5.86
5.96
6.04
6.09
6.13
6.16
6.19
12.5
25.0
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
AMERICAN OPTIONS
time
i
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
CALLABLE BONDS
12.1.3
431
The Option Adjusted Spread
Securities with embedded American options are often traded on an option-adjusted-spread
(OAS) basis. This spread reflects the discrepancy between the value of the security according to the model used to price it, and the traded value. For instance, on January 8,
2002, the August 2014, 7.5% callable bond discussed in Example 12.2 was trading for
V Q u ote (0, T ) = $146.5 (average bid and ask in Table 12.1). The term structure model
used to price this bond gave the price V0cb (T ) = $147.40, which is higher than the quoted
price. According to the model the market is undervaluing the callable bond, which suggests
a buy strategy. The OAS represents the increase in the spot curve that is necessary for the
model price V0cb (T ) to equal the traded price. A positive OAS implies that we had to
increase the yields of the model to match the traded price, suggesting that the bond will
generate a higher yield compared to the fair one even after accounting for the embedded
option.5 For instance, in Example 12.2 it turns out that if we increase the term structure in
a parallel fashion of only 11 basis points, and then redo the computations as in Table 12.3
(including fitting the interest rate tree) we obtain V0cb (T ) = $146.55, which equals the ask
price (see Table 12.1). Thus, in this case
Option Adjusted Spread = 11 basis points
There are numerous reasons why the OAS of a bond may be different from zero. For
instance, if a callable bond has low liquidity, it tends to trade at a discount compared to its
fair (option adjusted) value, yielding a positive OAS. In this case, the positive OAS may
reflect liquidity risk, that is, the risk of not being able to sell the bond for its fair value when
needed.
It is also important to realize that the size of the OAS depends on the model used to
compute it: It is possible that different term structure models may generate different OAS,
in the same way they generate different values of options even when using the same inputs,
as discussed in Chapter 11.
12.1.4
Dynamic Replication of Callable Bonds
The higher the option adjusted spread (OAS), the more underpriced is the bond compared
to the model, and thus the higher the incentive to purchase the bond should be. Indeed,
because of dynamic replication, if a callable bond turns out to be underpriced then one can
engage in the strategy “Buy Cheap and Sell Dear,” according to which a trader purchases
the underpriced callable bond, and sells a portfolio of non-callable bonds chosen in a way
it replicates the callable bonds. Dynamic replication was discussed in Chapters 9 and 10.
L
in a long-term
Recall in particular that the replication strategy involves a position Ni,j
S
security, such as a non-callable bond in this case, and a position Ni,j in the short-term
bond, expiring in period i + 1. There are two differences compared to the cases discussed
in Chapters 9 and 10:
1. The callable bond pays coupons, and therefore the dynamic strategy must take those
coupons into account.
5 Note
that a parallel shift in the spot curve results in an equal shift in the forward curve, so that the OAS can
alternatively be computed from a shift in the forward curve.
432
AMERICAN OPTIONS
2. The issuer should exercise optimally according to the tree, but it may not do so.
Therefore, the replicating strategy must take into account the possibility that the
issuer does not exercise when it should according to the model.
We use the simple Example 12.1 to illustrate the methodology.
EXAMPLE 12.3
Consider again Example 12.1. Assume that the market is trading this callable bond
at a positive OAS, and let for instance the price be P m k t (0, 3) = 100.1, while the
computations in Table 12.2 imply its price should be V0cb (3) = 100.3564. Let’s
assume for simplicity that a non-callable, 3% bond with maturity i = 3 is also
trading, and it is fairly priced according to the model. That is, from Panel A of Table
12.2 the non-callable bond is trading at P0 (3) = 100.5438.
Before discussing the arbitrage strategy, consider first the replicating portfolio.
Recall from Equations 9.9 and 9.10 in Chapter 9 that at any node (i, j) we must take
the following positions:
L
Ni,j
=
S
Ni,j
=
Vi+ 1,j − Vi+1,j +1
Pi+ 1,j (3) − Pi+1,j +1 (3)
$
1 #
L
Vi+ 1,j − Ni,j
× Pi+1,j (3)
100
(12.3)
(12.4)
where Vi,j is the value of the security we want to replicate (the callable bond),
and Pi,j (3) is the value of the security we use to replicate (the non-callable bond).
Equations 12.3 and 12.4 though do not take into account the fact that both Vi,j and
Pi,j pay coupons at each time i. The same logic discussed in Chapter 9 however
shows that with coupons, the only change is in Equation 12.4, which becomes
S
=
Ni,j
$
1 #
V
L
P
Vi+ 1,j + CFi+
1 − Ni,j × Pi+1,j (3) + CFi+1
100
(12.5)
V
P
where CFi+1
and CFi+
1 denote the cash flow at i + 1 paid by securities V and
P , respectively. For the case of the callable and non-callable bonds with identical
V
P
coupon rates, we have CFi+
1 = CFi+ 1 = 1.5.
As an example, we initially have (we use the notation u, uu to denote nodes, for
simplicity):
N0L
=
N0S
=
=
99.4667 − 100
V1,u − V1,d
=
= 0.5852
P1,u (3) − P1,d (3)
99.4667 − 100.3780
$
1 #
V1,u + CF1V − N0L × P1,u (3) + CF1P
100
1
[(99.4667 + 1.5) − 0.5852 × (99.4667 + 1.5)] = 0.4188
100
The initial value of this